  
  
                             [1XGAP - Reference Manual[101X
  
  
                           Release 4.6.4, 04-May-2013
  
  
                                 The GAP Group
  
  
  
  The GAP Group
      Email:    [7Xmailto:support@gap-system.org[107X
      Homepage: [7Xhttp://www.gap-system.org[107X
  
  -------------------------------------------------------
  [1XCopyright[101X
  [33X[0;0YCopyright  ©  (1987-2013)  for  the  core  part of the [5XGAP[105X system by the [5XGAP[105X
  Group.[133X
  
  [33X[0;0YMost  parts  of this distribution, including the core part of the [5XGAP[105X system
  are  distributed  under  the  terms  of  the GNU General Public License, see
  [7Xhttp://www.gnu.org/licenses/gpl.html[107X or the file [11XGPL[111X in the [11Xetc[111X directory of
  the [5XGAP[105X installation.[133X
  
  [33X[0;0YMore  detailed  information  about  copyright  and licenses of parts of this
  distribution can be found in Section [14X1.4[114X of this manual.[133X
  
  [33X[0;0Y[5XGAP[105X  is  developed  over  a long time and has many authors and contributors.
  More detailed information can be found in Section [14X1.2[114X of this manual.[133X
  
  
  -------------------------------------------------------
  
  
  [1XContents (ref)[101X
  
  1 [33X[0;0YPreface[133X
    1.1 [33X[0;0YThe [5XGAP[105X System[133X
    1.2 [33X[0;0YAuthors and Maintainers[133X
    1.3 [33X[0;0YAcknowledgements[133X
    1.4 [33X[0;0YCopyright and License[133X
    1.5 [33X[0;0YFurther Information about [5XGAP[105X[133X
  2 [33X[0;0YThe Help System[133X
    2.1 [33X[0;0YInvoking the Help[133X
    2.2 [33X[0;0YBrowsing through the Sections[133X
    2.3 [33X[0;0YChanging the Help Viewer[133X
      2.3-1 SetHelpViewer
    2.4 [33X[0;0YThe Pager Command[133X
      2.4-1 Pager
  3 [33X[0;0YRunning GAP[133X
    3.1 [33X[0;0YCommand Line Options[133X
    3.2 [33X[0;0YThe gap.ini and gaprc files[133X
      3.2-1 [33X[0;0YThe gap.ini file[133X
      3.2-2 [33X[0;0YThe gaprc file[133X
      3.2-3 [33X[0;0YConfiguring User preferences[133X
      3.2-4 DeclareUserPreference
    3.3 [33X[0;0YSaving and Loading a Workspace[133X
      3.3-1 SaveWorkspace
    3.4 [33X[0;0YTesting for the System Architecture[133X
      3.4-1 ARCH_IS_UNIX
      3.4-2 ARCH_IS_MAC_OS_X
      3.4-3 ARCH_IS_WINDOWS
    3.5 [33X[0;0YGlobal Values that Control the [5XGAP[105X Session[133X
      3.5-1 GAPInfo
    3.6 [33X[0;0YColoring the Prompt and Input[133X
      3.6-1 ColorPrompt
  4 [33X[0;0YThe Programming Language[133X
    4.1 [33X[0;0YLanguage Overview[133X
    4.2 [33X[0;0YLexical Structure[133X
    4.3 [33X[0;0YSymbols[133X
    4.4 [33X[0;0YWhitespaces[133X
    4.5 [33X[0;0YKeywords[133X
    4.6 [33X[0;0YIdentifiers[133X
      4.6-1 IsValidIdentifier
    4.7 [33X[0;0YExpressions[133X
    4.8 [33X[0;0YVariables[133X
      4.8-1 IsBound
      4.8-2 Unbind
    4.9 [33X[0;0YMore About Global Variables[133X
      4.9-1 IsReadOnlyGlobal
      4.9-2 MakeReadOnlyGlobal
      4.9-3 MakeReadWriteGlobal
      4.9-4 ValueGlobal
      4.9-5 IsBoundGlobal
      4.9-6 UnbindGlobal
      4.9-7 BindGlobal
      4.9-8 NamesGVars
      4.9-9 NamesSystemGVars
      4.9-10 NamesUserGVars
      4.9-11 TemporaryGlobalVarName
    4.10 [33X[0;0YNamespaces for [5XGAP[105X packages[133X
    4.11 [33X[0;0YFunction Calls[133X
      4.11-1 [33X[0;0YFunction Call With Arguments[133X
      4.11-2 [33X[0;0YFunction Call With Options[133X
    4.12 [33X[0;0YComparisons[133X
    4.13 [33X[0;0YArithmetic Operators[133X
    4.14 [33X[0;0YStatements[133X
    4.15 [33X[0;0YAssignments[133X
    4.16 [33X[0;0YProcedure Calls[133X
    4.17 [33X[0;0YIf[133X
    4.18 [33X[0;0YWhile[133X
    4.19 [33X[0;0YRepeat[133X
    4.20 [33X[0;0YFor[133X
    4.21 [33X[0;0YBreak[133X
    4.22 [33X[0;0YContinue[133X
    4.23 [33X[0;0YFunction[133X
    4.24 [33X[0;0YReturn (With or without Value)[133X
  5 [33X[0;0YFunctions[133X
    5.1 [33X[0;0YInformation about a function[133X
      5.1-1 NameFunction
      5.1-2 NumberArgumentsFunction
      5.1-3 NamesLocalVariablesFunction
      5.1-4 FilenameFunc
      5.1-5 StartlineFunc
      5.1-6 PageSource
    5.2 [33X[0;0YCalling a function with a list argument that is interpreted as several
    arguments[133X
      5.2-1 CallFuncList
    5.3 [33X[0;0YFunctions that do nothing[133X
      5.3-1 ReturnTrue
      5.3-2 ReturnFalse
      5.3-3 ReturnFail
      5.3-4 IdFunc
    5.4 [33X[0;0YFunction Types[133X
      5.4-1 IsFunction
      5.4-2 IsOperation
      5.4-3 FunctionsFamily
    5.5 [33X[0;0YNaming Conventions[133X
  6 [33X[0;0YMain Loop and Break Loop[133X
    6.1 [33X[0;0YMain Loop[133X
    6.2 [33X[0;0YSpecial Rules for Input Lines[133X
    6.3 [33X[0;0YView and Print[133X
      6.3-1 [33X[0;0YDefault delegations in the library[133X
      6.3-2 [33X[0;0YRecommendations for the implementation[133X
      6.3-3 View
      6.3-4 Print
      6.3-5 ViewObj
      6.3-6 Display
      6.3-7 SetNameObject
    6.4 [33X[0;0YBreak Loops[133X
      6.4-1 [33X[0;0Yquit from a break loop[133X
      6.4-2 [33X[0;0Yreturn from a break loop[133X
      6.4-3 OnBreak
      6.4-4 OnBreakMessage
      6.4-5 Where
    6.5 [33X[0;0YVariable Access in a Break Loop[133X
      6.5-1 [33X[0;0YDownEnv and UpEnv[133X
    6.6 [33X[0;0YError and ErrorCount[133X
      6.6-1 Error
      6.6-2 ErrorCount
    6.7 [33X[0;0YLeaving GAP[133X
      6.7-1 QUIT
      6.7-2 InstallAtExit
      6.7-3 SaveOnExitFile
    6.8 [33X[0;0YLine Editing[133X
    6.9 [33X[0;0YEditing using the [10Xreadline[110X library[133X
      6.9-1 [33X[0;0YReadline customization[133X
      6.9-2 [33X[0;0YThe command line history[133X
      6.9-3 SaveCommandLineHistory
      6.9-4 [33X[0;0YWriting your own command line editing functions[133X
    6.10 [33X[0;0YEditing Files[133X
      6.10-1 Edit
    6.11 [33X[0;0YEditor Support[133X
    6.12 [33X[0;0YChanging the Screen Size[133X
      6.12-1 SizeScreen
    6.13 [33X[0;0YTeaching Mode[133X
      6.13-1 TeachingMode
  7 [33X[0;0YDebugging and Profiling Facilities[133X
    7.1 [33X[0;0YRecovery from NoMethodFound-Errors[133X
      7.1-1 ShowArguments
      7.1-2 ShowArgument
      7.1-3 ShowDetails
      7.1-4 ShowMethods
      7.1-5 ShowOtherMethods
    7.2 [33X[0;0YInspecting Applicable Methods[133X
      7.2-1 ApplicableMethod
    7.3 [33X[0;0YTracing Methods[133X
      7.3-1 TraceMethods
      7.3-2 UntraceMethods
      7.3-3 TraceImmediateMethods
    7.4 [33X[0;0YInfo Functions[133X
      7.4-1 NewInfoClass
      7.4-2 DeclareInfoClass
      7.4-3 SetInfoLevel
      7.4-4 InfoLevel
      7.4-5 Info
      7.4-6 [33X[0;0YCustomizing [2XInfo[102X ([14X7.4-5[114X) statements[133X
      7.4-7 InfoWarning
    7.5 [33X[0;0YAssertions[133X
      7.5-1 SetAssertionLevel
      7.5-2 AssertionLevel
      7.5-3 Assert
    7.6 [33X[0;0YTiming[133X
      7.6-1 Runtimes
      7.6-2 Runtime
      7.6-3 time
    7.7 [33X[0;0YProfiling[133X
      7.7-1 ProfileGlobalFunctions
      7.7-2 ProfileOperations
      7.7-3 ProfileOperationsAndMethods
      7.7-4 ProfileFunctions
      7.7-5 UnprofileFunctions
      7.7-6 ProfileMethods
      7.7-7 UnprofileMethods
      7.7-8 DisplayProfile
      7.7-9 ClearProfile
      7.7-10 [33X[0;0YAn Example of Profiling[133X
      7.7-11 DisplayCacheStats
      7.7-12 ClearCacheStats
    7.8 [33X[0;0YInformation about the version used[133X
    7.9 [33X[0;0YTest Files[133X
      7.9-1 ReadTest
      7.9-2 [33X[0;0YStarting and stopping test[133X
      7.9-3 Test
    7.10 [33X[0;0YDebugging Recursion[133X
      7.10-1 SetRecursionTrapInterval
    7.11 [33X[0;0YGlobal Memory Information[133X
      7.11-1 GasmanStatistics
      7.11-2 GasmanMessageStatus
      7.11-3 GasmanLimits
  8 [33X[0;0YOptions Stack[133X
    8.1 [33X[0;0YFunctions Dealing with the Options Stack[133X
      8.1-1 PushOptions
      8.1-2 PopOptions
      8.1-3 ResetOptionsStack
      8.1-4 OnQuit
      8.1-5 ValueOption
      8.1-6 DisplayOptionsStack
      8.1-7 InfoOptions
    8.2 [33X[0;0YOptions Stack – an Example[133X
  9 [33X[0;0YFiles and Filenames[133X
    9.1 [33X[0;0YPortability[133X
      9.1-1 LastSystemError
    9.2 [33X[0;0YGAP Root Directories[133X
    9.3 [33X[0;0YDirectories[133X
      9.3-1 IsDirectory
      9.3-2 Directory
      9.3-3 DirectoryTemporary
      9.3-4 DirectoryCurrent
      9.3-5 DirectoriesLibrary
      9.3-6 DirectoriesSystemPrograms
      9.3-7 DirectoryContents
      9.3-8 DirectoryDesktop
      9.3-9 DirectoryHome
    9.4 [33X[0;0YFile Names[133X
      9.4-1 [33X[0;0YFilename[133X
    9.5 [33X[0;0YSpecial Filenames[133X
    9.6 [33X[0;0YFile Access[133X
      9.6-1 IsExistingFile
      9.6-2 IsReadableFile
      9.6-3 IsWritableFile
      9.6-4 IsExecutableFile
      9.6-5 IsDirectoryPath
    9.7 [33X[0;0YFile Operations[133X
      9.7-1 Read
      9.7-2 ReadAsFunction
      9.7-3 [33X[0;0YPrintTo and AppendTo[133X
      9.7-4 [33X[0;0YLogTo[133X
      9.7-5 [33X[0;0YInputLogTo[133X
      9.7-6 [33X[0;0YOutputLogTo[133X
      9.7-7 CrcFile
      9.7-8 RemoveFile
      9.7-9 Reread
  10 [33X[0;0YStreams[133X
    10.1 [33X[0;0YCategories for Streams and the StreamsFamily[133X
      10.1-1 IsStream
      10.1-2 IsClosedStream
      10.1-3 IsInputStream
      10.1-4 IsInputTextStream
      10.1-5 IsInputTextNone
      10.1-6 IsOutputStream
      10.1-7 IsOutputTextStream
      10.1-8 IsOutputTextNone
      10.1-9 StreamsFamily
    10.2 [33X[0;0YOperations applicable to All Streams[133X
      10.2-1 CloseStream
      10.2-2 FileDescriptorOfStream
      10.2-3 UNIXSelect
    10.3 [33X[0;0YOperations for Input Streams[133X
      10.3-1 Read
      10.3-2 ReadAsFunction
      10.3-3 ReadTest
      10.3-4 ReadByte
      10.3-5 ReadLine
      10.3-6 ReadAll
      10.3-7 IsEndOfStream
      10.3-8 PositionStream
      10.3-9 RewindStream
      10.3-10 SeekPositionStream
    10.4 [33X[0;0YOperations for Output Streams[133X
      10.4-1 WriteByte
      10.4-2 WriteLine
      10.4-3 WriteAll
      10.4-4 [33X[0;0YPrintTo and AppendTo (for streams)[133X
      10.4-5 LogTo
      10.4-6 InputLogTo
      10.4-7 OutputLogTo
      10.4-8 SetPrintFormattingStatus
    10.5 [33X[0;0YFile Streams[133X
      10.5-1 InputTextFile
      10.5-2 OutputTextFile
    10.6 [33X[0;0YUser Streams[133X
      10.6-1 InputTextUser
      10.6-2 OutputTextUser
      10.6-3 InputFromUser
    10.7 [33X[0;0YString Streams[133X
      10.7-1 InputTextString
      10.7-2 OutputTextString
    10.8 [33X[0;0YInput-Output Streams[133X
      10.8-1 IsInputOutputStream
      10.8-2 InputOutputLocalProcess
      10.8-3 ReadAllLine
    10.9 [33X[0;0YDummy Streams[133X
      10.9-1 InputTextNone
      10.9-2 OutputTextNone
    10.10 [33X[0;0YHandling of Streams in the Background[133X
      10.10-1 InstallCharReadHookFunc
      10.10-2 UnInstallCharReadHookFunc
    10.11 [33X[0;0YComma separated files[133X
      10.11-1 ReadCSV
      10.11-2 PrintCSV
  11 [33X[0;0YProcesses[133X
    11.1 [33X[0;0YProcess and Exec[133X
      11.1-1 Process
      11.1-2 Exec
  12 [33X[0;0YObjects and Elements[133X
    12.1 [33X[0;0YObjects[133X
      12.1-1 IsObject
    12.2 [33X[0;0YElements as equivalence classes[133X
    12.3 [33X[0;0YSets[133X
    12.4 [33X[0;0YDomains[133X
    12.5 [33X[0;0YIdentical Objects[133X
      12.5-1 IsIdenticalObj
      12.5-2 IsNotIdenticalObj
    12.6 [33X[0;0YMutability and Copyability[133X
      12.6-1 IsCopyable
      12.6-2 IsMutable
      12.6-3 Immutable
      12.6-4 MakeImmutable
      12.6-5 [33X[0;0YMutability of Iterators[133X
      12.6-6 [33X[0;0YMutability of Results of Arithmetic Operations[133X
    12.7 [33X[0;0YDuplication of Objects[133X
      12.7-1 ShallowCopy
      12.7-2 StructuralCopy
    12.8 [33X[0;0YOther Operations Applicable to any Object[133X
      12.8-1 SetName
      12.8-2 Name
      12.8-3 IsInternallyConsistent
      12.8-4 MemoryUsage
  13 [33X[0;0YTypes of Objects[133X
    13.1 [33X[0;0YFamilies[133X
      13.1-1 FamilyObj
    13.2 [33X[0;0YFilters[133X
      13.2-1 RankFilter
      13.2-2 NamesFilter
      13.2-3 ShowImpliedFilters
    13.3 [33X[0;0YCategories[133X
      13.3-1 CategoriesOfObject
    13.4 [33X[0;0YRepresentation[133X
      13.4-1 RepresentationsOfObject
    13.5 [33X[0;0YAttributes[133X
      13.5-1 KnownAttributesOfObject
    13.6 [33X[0;0YSetter and Tester for Attributes[133X
      13.6-1 Tester
      13.6-2 Setter
      13.6-3 AttributeValueNotSet
      13.6-4 InfoAttributes
      13.6-5 DisableAttributeValueStoring
      13.6-6 EnableAttributeValueStoring
    13.7 [33X[0;0YProperties[133X
      13.7-1 KnownPropertiesOfObject
      13.7-2 KnownTruePropertiesOfObject
    13.8 [33X[0;0YOther Filters[133X
    13.9 [33X[0;0YTypes[133X
      13.9-1 TypeObj
      13.9-2 DataType
  14 [33X[0;0YIntegers[133X
    14.1 [33X[0;0YIntegers: Global Variables[133X
      14.1-1 Integers
      14.1-2 IsIntegers
    14.2 [33X[0;0YElementary Operations for Integers[133X
      14.2-1 IsInt
      14.2-2 IsPosInt
      14.2-3 Int
      14.2-4 IsEvenInt
      14.2-5 IsOddInt
      14.2-6 AbsInt
      14.2-7 SignInt
      14.2-8 LogInt
      14.2-9 RootInt
      14.2-10 SmallestRootInt
      14.2-11 ListOfDigits
      14.2-12 Random
    14.3 [33X[0;0YQuotients and Remainders[133X
      14.3-1 QuoInt
      14.3-2 BestQuoInt
      14.3-3 RemInt
      14.3-4 GcdInt
      14.3-5 Gcdex
      14.3-6 LcmInt
      14.3-7 CoefficientsQadic
      14.3-8 CoefficientsMultiadic
      14.3-9 ChineseRem
      14.3-10 PowerModInt
    14.4 [33X[0;0YPrime Integers and Factorization[133X
      14.4-1 Primes
      14.4-2 IsPrimeInt
      14.4-3 PrimalityProof
      14.4-4 IsPrimePowerInt
      14.4-5 NextPrimeInt
      14.4-6 PrevPrimeInt
      14.4-7 FactorsInt
      14.4-8 PrimeDivisors
      14.4-9 PartialFactorization
      14.4-10 PrintFactorsInt
      14.4-11 PrimePowersInt
      14.4-12 DivisorsInt
    14.5 [33X[0;0YResidue Class Rings[133X
      14.5-1 \mod
      14.5-2 ZmodnZ
      14.5-3 ZmodnZObj
      14.5-4 IsZmodnZObj
    14.6 [33X[0;0YCheck Digits[133X
      14.6-1 CheckDigitISBN
      14.6-2 CheckDigitTestFunction
    14.7 [33X[0;0YRandom Sources[133X
      14.7-1 IsRandomSource
      14.7-2 Random
      14.7-3 State
      14.7-4 IsMersenneTwister
      14.7-5 RandomSource
  15 [33X[0;0YNumber Theory[133X
    15.1 [33X[0;0YInfoNumtheor (Info Class)[133X
      15.1-1 InfoNumtheor
    15.2 [33X[0;0YPrime Residues[133X
      15.2-1 PrimeResidues
      15.2-2 Phi
      15.2-3 Lambda
      15.2-4 GeneratorsPrimeResidues
    15.3 [33X[0;0YPrimitive Roots and Discrete Logarithms[133X
      15.3-1 OrderMod
      15.3-2 LogMod
      15.3-3 PrimitiveRootMod
      15.3-4 IsPrimitiveRootMod
    15.4 [33X[0;0YRoots Modulo Integers[133X
      15.4-1 Jacobi
      15.4-2 Legendre
      15.4-3 RootMod
      15.4-4 RootsMod
      15.4-5 RootsUnityMod
    15.5 [33X[0;0YMultiplicative Arithmetic Functions[133X
      15.5-1 Sigma
      15.5-2 Tau
      15.5-3 MoebiusMu
    15.6 [33X[0;0YContinued Fractions[133X
      15.6-1 ContinuedFractionExpansionOfRoot
      15.6-2 ContinuedFractionApproximationOfRoot
    15.7 [33X[0;0YMiscellaneous[133X
      15.7-1 TwoSquares
  16 [33X[0;0YCombinatorics[133X
    16.1 [33X[0;0YCombinatorial Numbers[133X
      16.1-1 Factorial
      16.1-2 Binomial
      16.1-3 Bell
      16.1-4 Bernoulli
      16.1-5 Stirling1
      16.1-6 Stirling2
    16.2 [33X[0;0YCombinations, Arrangements and Tuples[133X
      16.2-1 Combinations
      16.2-2 [33X[0;0YIterator and enumerator of combinations[133X
      16.2-3 NrCombinations
      16.2-4 Arrangements
      16.2-5 NrArrangements
      16.2-6 UnorderedTuples
      16.2-7 NrUnorderedTuples
      16.2-8 Tuples
      16.2-9 EnumeratorOfTuples
      16.2-10 IteratorOfTuples
      16.2-11 NrTuples
      16.2-12 PermutationsList
      16.2-13 NrPermutationsList
      16.2-14 Derangements
      16.2-15 NrDerangements
      16.2-16 PartitionsSet
      16.2-17 NrPartitionsSet
      16.2-18 Partitions
      16.2-19 IteratorOfPartitions
      16.2-20 NrPartitions
      16.2-21 OrderedPartitions
      16.2-22 NrOrderedPartitions
      16.2-23 PartitionsGreatestLE
      16.2-24 PartitionsGreatestEQ
      16.2-25 RestrictedPartitions
      16.2-26 NrRestrictedPartitions
      16.2-27 SignPartition
      16.2-28 AssociatedPartition
      16.2-29 PowerPartition
      16.2-30 PartitionTuples
      16.2-31 NrPartitionTuples
    16.3 [33X[0;0YFibonacci and Lucas Sequences[133X
      16.3-1 Fibonacci
      16.3-2 Lucas
    16.4 [33X[0;0YPermanent of a Matrix[133X
      16.4-1 Permanent
  17 [33X[0;0YRational Numbers[133X
    17.1 [33X[0;0YRationals: Global Variables[133X
      17.1-1 Rationals
    17.2 [33X[0;0YElementary Operations for Rationals[133X
      17.2-1 IsRat
      17.2-2 IsPosRat
      17.2-3 IsNegRat
      17.2-4 NumeratorRat
      17.2-5 DenominatorRat
      17.2-6 Rat
      17.2-7 Random
  18 [33X[0;0YCyclotomic Numbers[133X
    18.1 [33X[0;0YOperations for Cyclotomics[133X
      18.1-1 E
      18.1-2 Cyclotomics
      18.1-3 IsCyclotomic
      18.1-4 IsIntegralCyclotomic
      18.1-5 Int
      18.1-6 String
      18.1-7 Conductor
      18.1-8 AbsoluteValue
      18.1-9 RoundCyc
      18.1-10 CoeffsCyc
      18.1-11 DenominatorCyc
      18.1-12 ExtRepOfObj
      18.1-13 DescriptionOfRootOfUnity
      18.1-14 IsGaussInt
      18.1-15 IsGaussRat
      18.1-16 DefaultField
    18.2 [33X[0;0YInfinity[133X
      18.2-1 IsInfinity
    18.3 [33X[0;0YComparisons of Cyclotomics[133X
    18.4 [33X[0;0YATLAS Irrationalities[133X
      18.4-1 [33X[0;0YEB, EC, [22X...[122X, EH[133X
      18.4-2 [33X[0;0YEI and ER[133X
      18.4-3 [33X[0;0YEY, EX, [22X...[122X, ES[133X
      18.4-4 [33X[0;0YEM, EL, [22X...[122X, EJ[133X
      18.4-5 NK
      18.4-6 AtlasIrrationality
    18.5 [33X[0;0YGalois Conjugacy of Cyclotomics[133X
      18.5-1 GaloisCyc
      18.5-2 ComplexConjugate
      18.5-3 StarCyc
      18.5-4 Quadratic
      18.5-5 GaloisMat
      18.5-6 RationalizedMat
    18.6 [33X[0;0YInternally Represented Cyclotomics[133X
      18.6-1 SetCyclotomicsLimit
  19 [33X[0;0YFloats[133X
    19.1 [33X[0;0YA sample run[133X
    19.2 [33X[0;0YMethods[133X
      19.2-1 [33X[0;0YMathematical operations[133X
      19.2-2 EqFloat
      19.2-3 PrecisionFloat
      19.2-4 [33X[0;0YInterval operations[133X
      19.2-5 IsPInfinity
      19.2-6 FLOAT
      19.2-7 Float
      19.2-8 Rat
      19.2-9 SetFloats
    19.3 [33X[0;0YHigh-precision-specific methods[133X
    19.4 [33X[0;0YComplex arithmetic[133X
    19.5 [33X[0;0YInterval-specific methods[133X
  20 [33X[0;0YBooleans[133X
    20.1 [33X[0;0YIsBool (Filter)[133X
      20.1-1 IsBool
    20.2 [33X[0;0YFail (Variable)[133X
      20.2-1 fail
    20.3 [33X[0;0YComparisons of Booleans[133X
      20.3-1 [33X[0;0YEquality and inequality of Booleans[133X
      20.3-2 [33X[0;0YOrdering of Booleans[133X
    20.4 [33X[0;0YOperations for Booleans[133X
      20.4-1 [33X[0;0YLogical disjunction[133X
      20.4-2 [33X[0;0YLogical conjunction[133X
      20.4-3 [33X[0;0YLogical negation[133X
  21 [33X[0;0YLists[133X
    21.1 [33X[0;0YList Categories[133X
      21.1-1 IsList
      21.1-2 IsDenseList
      21.1-3 IsHomogeneousList
      21.1-4 IsTable
      21.1-5 IsRectangularTable
      21.1-6 IsConstantTimeAccessList
    21.2 [33X[0;0YBasic Operations for Lists[133X
      21.2-1 \[\]
    21.3 [33X[0;0YList Elements[133X
      21.3-1 \{\}
    21.4 [33X[0;0YList Assignment[133X
      21.4-1 \{\}\:\=
      21.4-2 Add
      21.4-3 Remove
      21.4-4 CopyListEntries
      21.4-5 Append
    21.5 [33X[0;0YIsBound and Unbind for Lists[133X
      21.5-1 IsBound
      21.5-2 Unbind
    21.6 [33X[0;0YIdentical Lists[133X
    21.7 [33X[0;0YDuplication of Lists[133X
    21.8 [33X[0;0YMembership Test for Lists[133X
      21.8-1 \in
    21.9 [33X[0;0YEnlarging Internally Represented Lists[133X
      21.9-1 EmptyPlist
    21.10 [33X[0;0YComparisons of Lists[133X
    21.11 [33X[0;0YArithmetic for Lists[133X
    21.12 [33X[0;0YFilters Controlling the Arithmetic Behaviour of Lists[133X
      21.12-1 IsGeneralizedRowVector
      21.12-2 IsMultiplicativeGeneralizedRowVector
      21.12-3 IsListDefault
      21.12-4 NestingDepthA
      21.12-5 NestingDepthM
    21.13 [33X[0;0YAdditive Arithmetic for Lists[133X
      21.13-1 [33X[0;0YZero for lists[133X
      21.13-2 [33X[0;0YAdditiveInverse for lists[133X
      21.13-3 [33X[0;0YAddition of lists[133X
      21.13-4 [33X[0;0YSubtraction of lists[133X
    21.14 [33X[0;0YMultiplicative Arithmetic for Lists[133X
      21.14-1 [33X[0;0YOne for lists[133X
      21.14-2 [33X[0;0YInverse for lists[133X
      21.14-3 [33X[0;0YMultiplication of lists[133X
      21.14-4 [33X[0;0YDivision of lists[133X
      21.14-5 [33X[0;0Ymod for lists[133X
      21.14-6 [33X[0;0YLeft quotients of lists[133X
    21.15 [33X[0;0YMutability Status and List Arithmetic[133X
      21.15-1 ListWithIdenticalEntries
    21.16 [33X[0;0YFinding Positions in Lists[133X
      21.16-1 Position
      21.16-2 Positions
      21.16-3 PositionCanonical
      21.16-4 PositionNthOccurrence
      21.16-5 PositionSorted
      21.16-6 PositionSet
      21.16-7 PositionProperty
      21.16-8 PositionsProperty
      21.16-9 PositionBound
      21.16-10 PositionNot
      21.16-11 PositionNonZero
      21.16-12 PositionSublist
      21.16-13 PositionFirstComponent
    21.17 [33X[0;0YProperties and Attributes for Lists[133X
      21.17-1 IsMatchingSublist
      21.17-2 IsDuplicateFree
      21.17-3 IsSortedList
      21.17-4 IsSSortedList
      21.17-5 Length
      21.17-6 ConstantTimeAccessList
    21.18 [33X[0;0YSorting Lists[133X
      21.18-1 Sort
      21.18-2 SortParallel
      21.18-3 Sortex
      21.18-4 SortingPerm
    21.19 [33X[0;0YSorted Lists and Sets[133X
      21.19-1 \in
      21.19-2 IsEqualSet
      21.19-3 IsSubsetSet
      21.19-4 AddSet
      21.19-5 RemoveSet
      21.19-6 UniteSet
      21.19-7 IntersectSet
      21.19-8 SubtractSet
    21.20 [33X[0;0YOperations for Lists[133X
      21.20-1 Concatenation
      21.20-2 Compacted
      21.20-3 Collected
      21.20-4 DuplicateFreeList
      21.20-5 AsDuplicateFreeList
      21.20-6 Flat
      21.20-7 Reversed
      21.20-8 Shuffle
      21.20-9 IsLexicographicallyLess
      21.20-10 Apply
      21.20-11 Perform
      21.20-12 PermListList
      21.20-13 [33X[0;0YMaximum[133X
      21.20-14 [33X[0;0YMinimum[133X
      21.20-15 [33X[0;0YMaximumList and MinimumList[133X
      21.20-16 [33X[0;0YCartesian[133X
      21.20-17 [33X[0;0YIteratorOfCartesianProduct[133X
      21.20-18 Permuted
      21.20-19 List
      21.20-20 Filtered
      21.20-21 Number
      21.20-22 First
      21.20-23 ForAll
      21.20-24 ForAny
      21.20-25 Product
      21.20-26 Sum
      21.20-27 Iterated
      21.20-28 ListN
    21.21 [33X[0;0YAdvanced List Manipulations[133X
      21.21-1 ListX
      21.21-2 SetX
      21.21-3 SumX
      21.21-4 ProductX
    21.22 [33X[0;0YRanges[133X
      21.22-1 IsRange
      21.22-2 ConvertToRangeRep
    21.23 [33X[0;0YEnumerators[133X
      21.23-1 IsQuickPositionList
  22 [33X[0;0YBoolean Lists[133X
    22.1 [33X[0;0YIsBlist (Filter)[133X
      22.1-1 IsBlist
    22.2 [33X[0;0YBoolean Lists Representing Subsets[133X
      22.2-1 BlistList
      22.2-2 ListBlist
      22.2-3 SizeBlist
      22.2-4 IsSubsetBlist
    22.3 [33X[0;0YSet Operations via Boolean Lists[133X
      22.3-1 [33X[0;0YUnionBlist[133X
      22.3-2 [33X[0;0YIntersectionBlist[133X
      22.3-3 DifferenceBlist
    22.4 [33X[0;0YFunction that Modify Boolean Lists[133X
      22.4-1 UniteBlist
      22.4-2 UniteBlistList
      22.4-3 IntersectBlist
      22.4-4 SubtractBlist
    22.5 [33X[0;0YMore about Boolean Lists[133X
      22.5-1 IsBlistRep
  23 [33X[0;0YRow Vectors[133X
    23.1 [33X[0;0YIsRowVector (Filter)[133X
      23.1-1 IsRowVector
    23.2 [33X[0;0YOperators for Row Vectors[133X
      23.2-1 NormedRowVector
    23.3 [33X[0;0YRow Vectors over Finite Fields[133X
      23.3-1 [33X[0;0YConvertToVectorRep[133X
      23.3-2 NumberFFVector
    23.4 [33X[0;0YCoefficient List Arithmetic[133X
      23.4-1 AddRowVector
      23.4-2 AddCoeffs
      23.4-3 MultRowVector
      23.4-4 CoeffsMod
    23.5 [33X[0;0YShifting and Trimming Coefficient Lists[133X
      23.5-1 LeftShiftRowVector
      23.5-2 RightShiftRowVector
      23.5-3 ShrinkRowVector
      23.5-4 RemoveOuterCoeffs
    23.6 [33X[0;0YFunctions for Coding Theory[133X
      23.6-1 WeightVecFFE
      23.6-2 DistanceVecFFE
      23.6-3 DistancesDistributionVecFFEsVecFFE
      23.6-4 DistancesDistributionMatFFEVecFFE
      23.6-5 AClosestVectorCombinationsMatFFEVecFFE
      23.6-6 CosetLeadersMatFFE
    23.7 [33X[0;0YVectors as coefficients of polynomials[133X
      23.7-1 ValuePol
      23.7-2 ProductCoeffs
      23.7-3 ReduceCoeffs
      23.7-4 ReduceCoeffsMod
      23.7-5 PowerModCoeffs
      23.7-6 ShiftedCoeffs
  24 [33X[0;0YMatrices[133X
    24.1 [33X[0;0YInfoMatrix (Info Class)[133X
      24.1-1 InfoMatrix
    24.2 [33X[0;0YCategories of Matrices[133X
      24.2-1 IsMatrix
      24.2-2 IsOrdinaryMatrix
      24.2-3 IsLieMatrix
    24.3 [33X[0;0YOperators for Matrices[133X
    24.4 [33X[0;0YProperties and Attributes of Matrices[133X
      24.4-1 DimensionsMat
      24.4-2 DefaultFieldOfMatrix
      24.4-3 TraceMat
      24.4-4 DeterminantMat
      24.4-5 DeterminantMatDestructive
      24.4-6 DeterminantMatDivFree
      24.4-7 IsMonomialMatrix
      24.4-8 IsDiagonalMat
      24.4-9 IsUpperTriangularMat
      24.4-10 IsLowerTriangularMat
    24.5 [33X[0;0YMatrix Constructions[133X
      24.5-1 IdentityMat
      24.5-2 NullMat
      24.5-3 EmptyMatrix
      24.5-4 DiagonalMat
      24.5-5 PermutationMat
      24.5-6 TransposedMatImmutable
      24.5-7 TransposedMatDestructive
      24.5-8 KroneckerProduct
      24.5-9 ReflectionMat
      24.5-10 PrintArray
    24.6 [33X[0;0YRandom Matrices[133X
      24.6-1 RandomMat
      24.6-2 RandomInvertibleMat
      24.6-3 RandomUnimodularMat
    24.7 [33X[0;0YMatrices Representing Linear Equations and the Gaussian Algorithm[133X
      24.7-1 RankMat
      24.7-2 TriangulizedMat
      24.7-3 TriangulizeMat
      24.7-4 NullspaceMat
      24.7-5 NullspaceMatDestructive
      24.7-6 SolutionMat
      24.7-7 SolutionMatDestructive
      24.7-8 BaseFixedSpace
    24.8 [33X[0;0YEigenvectors and eigenvalues[133X
      24.8-1 GeneralisedEigenvalues
      24.8-2 GeneralisedEigenspaces
      24.8-3 Eigenvalues
      24.8-4 Eigenspaces
      24.8-5 Eigenvectors
    24.9 [33X[0;0YElementary Divisors[133X
      24.9-1 ElementaryDivisorsMat
      24.9-2 ElementaryDivisorsTransformationsMat
      24.9-3 DiagonalizeMat
    24.10 [33X[0;0YEchelonized Matrices[133X
      24.10-1 SemiEchelonMat
      24.10-2 SemiEchelonMatDestructive
      24.10-3 SemiEchelonMatTransformation
      24.10-4 SemiEchelonMats
      24.10-5 SemiEchelonMatsDestructive
    24.11 [33X[0;0YMatrices as Basis of a Row Space[133X
      24.11-1 BaseMat
      24.11-2 BaseMatDestructive
      24.11-3 BaseOrthogonalSpaceMat
      24.11-4 SumIntersectionMat
      24.11-5 BaseSteinitzVectors
    24.12 [33X[0;0YTriangular Matrices[133X
      24.12-1 DiagonalOfMat
      24.12-2 UpperSubdiagonal
      24.12-3 DepthOfUpperTriangularMatrix
    24.13 [33X[0;0YMatrices as Linear Mappings[133X
      24.13-1 CharacteristicPolynomial
      24.13-2 JordanDecomposition
      24.13-3 BlownUpMat
      24.13-4 BlownUpVector
      24.13-5 CompanionMat
    24.14 [33X[0;0YMatrices over Finite Fields[133X
      24.14-1 ImmutableMatrix
      24.14-2 ConvertToMatrixRep
      24.14-3 ProjectiveOrder
      24.14-4 SimultaneousEigenvalues
    24.15 [33X[0;0YInverse and Nullspace of an Integer Matrix Modulo an Ideal[133X
      24.15-1 InverseMatMod
      24.15-2 NullspaceModQ
    24.16 [33X[0;0YSpecial Multiplication Algorithms for Matrices over GF(2)[133X
      24.16-1 PROD_GF2MAT_GF2MAT_SIMPLE
      24.16-2 PROD_GF2MAT_GF2MAT_ADVANCED
    24.17 [33X[0;0YBlock Matrices[133X
      24.17-1 AsBlockMatrix
      24.17-2 BlockMatrix
      24.17-3 MatrixByBlockMatrix
  25 [33X[0;0YIntegral matrices and lattices[133X
    25.1 [33X[0;0YLinear equations over the integers and Integral Matrices[133X
      25.1-1 NullspaceIntMat
      25.1-2 SolutionIntMat
      25.1-3 SolutionNullspaceIntMat
      25.1-4 BaseIntMat
      25.1-5 BaseIntersectionIntMats
      25.1-6 ComplementIntMat
    25.2 [33X[0;0YNormal Forms over the Integers[133X
      25.2-1 TriangulizedIntegerMat
      25.2-2 TriangulizedIntegerMatTransform
      25.2-3 TriangulizeIntegerMat
      25.2-4 HermiteNormalFormIntegerMat
      25.2-5 HermiteNormalFormIntegerMatTransform
      25.2-6 SmithNormalFormIntegerMat
      25.2-7 SmithNormalFormIntegerMatTransforms
      25.2-8 DiagonalizeIntMat
      25.2-9 NormalFormIntMat
      25.2-10 AbelianInvariantsOfList
    25.3 [33X[0;0YDeterminant of an integer matrix[133X
      25.3-1 DeterminantIntMat
    25.4 [33X[0;0YDecompositions[133X
      25.4-1 Decomposition
      25.4-2 LinearIndependentColumns
      25.4-3 PadicCoefficients
      25.4-4 IntegralizedMat
      25.4-5 DecompositionInt
    25.5 [33X[0;0YLattice Reduction[133X
      25.5-1 LLLReducedBasis
      25.5-2 LLLReducedGramMat
    25.6 [33X[0;0YOrthogonal Embeddings[133X
      25.6-1 OrthogonalEmbeddings
      25.6-2 ShortestVectors
  26 [33X[0;0YVector and matrix objects[133X
    26.1 [33X[0;0YFundamental ideas and rules[133X
    26.2 [33X[0;0YCategories of vectors and matrices[133X
    26.3 [33X[0;0YConstructing vector and matrix objects[133X
    26.4 [33X[0;0YOperations for row vector objects[133X
    26.5 [33X[0;0YOperations for row list matrix objects[133X
    26.6 [33X[0;0YOperations for flat matrix objects[133X
  27 [33X[0;0YStrings and Characters[133X
    27.1 [33X[0;0YIsChar and IsString[133X
      27.1-1 IsChar
      27.1-2 IsString
      27.1-3 [33X[0;0YStrings As Lists[133X
      27.1-4 [33X[0;0YPrinting Strings[133X
    27.2 [33X[0;0YSpecial Characters[133X
    27.3 [33X[0;0YInternally Represented Strings[133X
      27.3-1 IsStringRep
      27.3-2 ConvertToStringRep
      27.3-3 IsEmptyString
      27.3-4 EmptyString
      27.3-5 CharsFamily
    27.4 [33X[0;0YRecognizing Characters[133X
      27.4-1 IsDigitChar
      27.4-2 IsLowerAlphaChar
      27.4-3 IsUpperAlphaChar
      27.4-4 IsAlphaChar
    27.5 [33X[0;0YComparisons of Strings[133X
      27.5-1 \=
      27.5-2 \<
    27.6 [33X[0;0YOperations to Produce or Manipulate Strings[133X
      27.6-1 DisplayString
      27.6-2 DEFAULTDISPLAYSTRING
      27.6-3 ViewString
      27.6-4 DEFAULTVIEWSTRING
      27.6-5 PrintString
      27.6-6 String
      27.6-7 StripLineBreakCharacters
      27.6-8 HexStringInt
      27.6-9 StringPP
      27.6-10 WordAlp
      27.6-11 LowercaseString
      27.6-12 SplitString
      27.6-13 ReplacedString
      27.6-14 NormalizeWhitespace
      27.6-15 NormalizedWhitespace
      27.6-16 RemoveCharacters
      27.6-17 JoinStringsWithSeparator
      27.6-18 Chomp
      27.6-19 NumbersString
      27.6-20 StringNumbers
    27.7 [33X[0;0YCharacter Conversion[133X
      27.7-1 IntChar
      27.7-2 CharInt
      27.7-3 SIntChar
      27.7-4 CharSInt
    27.8 [33X[0;0YOperations to Evaluate Strings[133X
      27.8-1 Int
      27.8-2 Ordinal
      27.8-3 EvalString
      27.8-4 CrcString
    27.9 [33X[0;0YCalendar Arithmetic[133X
      27.9-1 DaysInYear
      27.9-2 DaysInMonth
      27.9-3 DMYDay
      27.9-4 DayDMY
      27.9-5 WeekDay
      27.9-6 StringDate
      27.9-7 HMSMSec
      27.9-8 SecHMSM
      27.9-9 StringTime
      27.9-10 SecondsDMYhms
      27.9-11 DMYhmsSeconds
    27.10 [33X[0;0YObtaining LaTeX Representations of Objects[133X
  28 [33X[0;0YDictionaries and General Hash Tables[133X
    28.1 [33X[0;0YUsing Dictionaries[133X
    28.2 [33X[0;0YDictionaries[133X
      28.2-1 NewDictionary
    28.3 [33X[0;0YDictionaries via Binary Lists[133X
      28.3-1 DictionaryByPosition
      28.3-2 IsDictionary
      28.3-3 IsLookupDictionary
      28.3-4 AddDictionary
      28.3-5 KnowsDictionary
      28.3-6 LookupDictionary
    28.4 [33X[0;0YGeneral Hash Tables[133X
    28.5 [33X[0;0YHash keys[133X
      28.5-1 DenseIntKey
      28.5-2 SparseIntKey
    28.6 [33X[0;0YDense hash tables[133X
      28.6-1 DenseHashTable
    28.7 [33X[0;0YSparse hash tables[133X
      28.7-1 SparseHashTable
      28.7-2 DoubleHashArraySize
  29 [33X[0;0YRecords[133X
    29.1 [33X[0;0YIsRecord and RecNames[133X
      29.1-1 IsRecord
      29.1-2 RecNames
    29.2 [33X[0;0YAccessing Record Elements[133X
    29.3 [33X[0;0YRecord Assignment[133X
    29.4 [33X[0;0YIdentical Records[133X
    29.5 [33X[0;0YComparisons of Records[133X
    29.6 [33X[0;0YIsBound and Unbind for Records[133X
      29.6-1 IsBound
      29.6-2 Unbind
    29.7 [33X[0;0YRecord Access Operations[133X
      29.7-1 NameRNam
      29.7-2 RNamObj
      29.7-3 \.
  30 [33X[0;0YCollections[133X
    30.1 [33X[0;0YIsCollection (Filter)[133X
      30.1-1 IsCollection
    30.2 [33X[0;0YCollection Families[133X
      30.2-1 CollectionsFamily
      30.2-2 IsCollectionFamily
      30.2-3 ElementsFamily
      30.2-4 CategoryCollections
    30.3 [33X[0;0YLists and Collections[133X
      30.3-1 IsListOrCollection
      30.3-2 Enumerator
      30.3-3 EnumeratorSorted
      30.3-4 EnumeratorByFunctions
      30.3-5 List
      30.3-6 SortedList
      30.3-7 SSortedList
      30.3-8 AsList
      30.3-9 AsSortedList
      30.3-10 AsSSortedList
      30.3-11 Elements
    30.4 [33X[0;0YAttributes and Properties for Collections[133X
      30.4-1 IsEmpty
      30.4-2 IsFinite
      30.4-3 IsTrivial
      30.4-4 IsNonTrivial
      30.4-5 IsWholeFamily
      30.4-6 Size
      30.4-7 Representative
      30.4-8 RepresentativeSmallest
    30.5 [33X[0;0YOperations for Collections[133X
      30.5-1 IsSubset
      30.5-2 [33X[0;0YIntersection[133X
      30.5-3 [33X[0;0YUnion[133X
      30.5-4 Difference
    30.6 [33X[0;0YMembership Test for Collections[133X
      30.6-1 \in
    30.7 [33X[0;0YRandom Elements[133X
      30.7-1 Random
      30.7-2 PseudoRandom
      30.7-3 RandomList
    30.8 [33X[0;0YIterators[133X
      30.8-1 Iterator
      30.8-2 IteratorSorted
      30.8-3 IsIterator
      30.8-4 IsDoneIterator
      30.8-5 NextIterator
      30.8-6 IteratorList
      30.8-7 TrivialIterator
      30.8-8 IteratorByFunctions
  31 [33X[0;0YDomains and their Elements[133X
    31.1 [33X[0;0YOperational Structure of Domains[133X
    31.2 [33X[0;0YEquality and Comparison of Domains[133X
    31.3 [33X[0;0YConstructing Domains[133X
    31.4 [33X[0;0YChanging the Structure[133X
    31.5 [33X[0;0YChanging the Representation[133X
    31.6 [33X[0;0YDomain Categories[133X
    31.7 [33X[0;0YParents[133X
      31.7-1 Parent
    31.8 [33X[0;0YConstructing Subdomains[133X
    31.9 [33X[0;0YOperations for Domains[133X
      31.9-1 IsGeneralizedDomain
      31.9-2 GeneratorsOfDomain
      31.9-3 Domain
    31.10 [33X[0;0YAttributes and Properties of Elements[133X
      31.10-1 Characteristic
      31.10-2 OneImmutable
      31.10-3 ZeroImmutable
      31.10-4 MultiplicativeZeroOp
      31.10-5 IsOne
      31.10-6 IsZero
      31.10-7 IsIdempotent
      31.10-8 InverseImmutable
      31.10-9 AdditiveInverseImmutable
      31.10-10 Order
    31.11 [33X[0;0YComparison Operations for Elements[133X
      31.11-1 [33X[0;0Y\= and \<[133X
      31.11-2 CanEasilyCompareElements
    31.12 [33X[0;0YArithmetic Operations for Elements[133X
      31.12-1 [33X[0;0Y\+, \*, \/, \^, \mod[133X
      31.12-2 LeftQuotient
      31.12-3 Comm
      31.12-4 LieBracket
      31.12-5 Sqrt
    31.13 [33X[0;0YRelations Between Domains[133X
      31.13-1 UseSubsetRelation
      31.13-2 UseFactorRelation
      31.13-3 UseIsomorphismRelation
      31.13-4 InstallSubsetMaintenance
      31.13-5 InstallFactorMaintenance
      31.13-6 InstallIsomorphismMaintenance
    31.14 [33X[0;0YUseful Categories of Elements[133X
      31.14-1 IsExtAElement
      31.14-2 IsNearAdditiveElement
      31.14-3 IsAdditiveElement
      31.14-4 IsNearAdditiveElementWithZero
      31.14-5 IsAdditiveElementWithZero
      31.14-6 IsNearAdditiveElementWithInverse
      31.14-7 IsAdditiveElementWithInverse
      31.14-8 IsExtLElement
      31.14-9 IsExtRElement
      31.14-10 IsMultiplicativeElement
      31.14-11 IsMultiplicativeElementWithOne
      31.14-12 IsMultiplicativeElementWithZero
      31.14-13 IsMultiplicativeElementWithInverse
      31.14-14 IsVector
      31.14-15 IsNearRingElement
      31.14-16 IsRingElement
      31.14-17 IsNearRingElementWithOne
      31.14-18 IsRingElementWithOne
      31.14-19 IsNearRingElementWithInverse
      31.14-20 IsRingElementWithInverse
    31.15 [33X[0;0YUseful Categories for all Elements of a Family[133X
      31.15-1 IsAssociativeElement
      31.15-2 IsAdditivelyCommutativeElement
      31.15-3 IsCommutativeElement
      31.15-4 IsFiniteOrderElement
      31.15-5 IsJacobianElement
      31.15-6 IsZeroSquaredElement
  32 [33X[0;0YMappings[133X
    32.1 [33X[0;0YIsDirectProductElement (Filter)[133X
      32.1-1 IsDirectProductElement
    32.2 [33X[0;0YCreating Mappings[133X
      32.2-1 GeneralMappingByElements
      32.2-2 [33X[0;0YMappingByFunction[133X
      32.2-3 InverseGeneralMapping
      32.2-4 CompositionMapping
      32.2-5 CompositionMapping2
      32.2-6 IsCompositionMappingRep
      32.2-7 ConstituentsCompositionMapping
      32.2-8 ZeroMapping
      32.2-9 IdentityMapping
      32.2-10 [33X[0;0YEmbedding[133X
      32.2-11 [33X[0;0YProjection[133X
      32.2-12 RestrictedMapping
    32.3 [33X[0;0YProperties and Attributes of (General) Mappings[133X
      32.3-1 IsTotal
      32.3-2 IsSingleValued
      32.3-3 IsMapping
      32.3-4 IsInjective
      32.3-5 IsSurjective
      32.3-6 IsBijective
      32.3-7 Range
      32.3-8 Source
      32.3-9 UnderlyingRelation
      32.3-10 UnderlyingGeneralMapping
    32.4 [33X[0;0YImages under Mappings[133X
      32.4-1 ImagesSource
      32.4-2 ImagesRepresentative
      32.4-3 ImagesElm
      32.4-4 ImagesSet
      32.4-5 ImageElm
      32.4-6 [33X[0;0YImage[133X
      32.4-7 [33X[0;0YImages[133X
    32.5 [33X[0;0YPreimages under Mappings[133X
      32.5-1 PreImagesRange
      32.5-2 PreImagesElm
      32.5-3 PreImageElm
      32.5-4 PreImagesRepresentative
      32.5-5 PreImagesSet
      32.5-6 [33X[0;0YPreImage[133X
      32.5-7 [33X[0;0YPreImages[133X
    32.6 [33X[0;0YArithmetic Operations for General Mappings[133X
    32.7 [33X[0;0YMappings which are Compatible with Algebraic Structures[133X
    32.8 [33X[0;0YMagma Homomorphisms[133X
      32.8-1 IsMagmaHomomorphism
      32.8-2 MagmaHomomorphismByFunctionNC
      32.8-3 NaturalHomomorphismByGenerators
    32.9 [33X[0;0YMappings that Respect Multiplication[133X
      32.9-1 RespectsMultiplication
      32.9-2 RespectsOne
      32.9-3 RespectsInverses
      32.9-4 IsGroupGeneralMapping
      32.9-5 KernelOfMultiplicativeGeneralMapping
      32.9-6 CoKernelOfMultiplicativeGeneralMapping
    32.10 [33X[0;0YMappings that Respect Addition[133X
      32.10-1 RespectsAddition
      32.10-2 RespectsAdditiveInverses
      32.10-3 RespectsZero
      32.10-4 IsAdditiveGroupGeneralMapping
      32.10-5 KernelOfAdditiveGeneralMapping
      32.10-6 CoKernelOfAdditiveGeneralMapping
    32.11 [33X[0;0YLinear Mappings[133X
      32.11-1 RespectsScalarMultiplication
      32.11-2 IsLeftModuleGeneralMapping
      32.11-3 IsLinearMapping
    32.12 [33X[0;0YRing Homomorphisms[133X
      32.12-1 IsRingGeneralMapping
      32.12-2 IsRingWithOneGeneralMapping
      32.12-3 IsAlgebraGeneralMapping
      32.12-4 IsAlgebraWithOneGeneralMapping
      32.12-5 IsFieldHomomorphism
    32.13 [33X[0;0YGeneral Mappings[133X
      32.13-1 IsGeneralMapping
      32.13-2 IsConstantTimeAccessGeneralMapping
      32.13-3 IsEndoGeneralMapping
    32.14 [33X[0;0YTechnical Matters Concerning General Mappings[133X
      32.14-1 IsSPGeneralMapping
      32.14-2 IsGeneralMappingFamily
      32.14-3 FamilyRange
      32.14-4 FamilySource
      32.14-5 FamiliesOfGeneralMappingsAndRanges
      32.14-6 GeneralMappingsFamily
      32.14-7 TypeOfDefaultGeneralMapping
  33 [33X[0;0YRelations[133X
    33.1 [33X[0;0YGeneral Binary Relations[133X
      33.1-1 IsBinaryRelation
      33.1-2 BinaryRelationByElements
      33.1-3 [33X[0;0YIdentityBinaryRelation[133X
      33.1-4 EmptyBinaryRelation
    33.2 [33X[0;0YProperties and Attributes of Binary Relations[133X
      33.2-1 IsReflexiveBinaryRelation
      33.2-2 IsSymmetricBinaryRelation
      33.2-3 IsTransitiveBinaryRelation
      33.2-4 IsAntisymmetricBinaryRelation
      33.2-5 IsPreOrderBinaryRelation
      33.2-6 IsPartialOrderBinaryRelation
      33.2-7 IsHasseDiagram
      33.2-8 IsEquivalenceRelation
      33.2-9 Successors
      33.2-10 DegreeOfBinaryRelation
      33.2-11 PartialOrderOfHasseDiagram
    33.3 [33X[0;0YBinary Relations on Points[133X
      33.3-1 BinaryRelationOnPoints
      33.3-2 RandomBinaryRelationOnPoints
      33.3-3 [33X[0;0YAsBinaryRelationOnPoints[133X
    33.4 [33X[0;0YClosure Operations and Other Constructors[133X
      33.4-1 ReflexiveClosureBinaryRelation
      33.4-2 SymmetricClosureBinaryRelation
      33.4-3 TransitiveClosureBinaryRelation
      33.4-4 HasseDiagramBinaryRelation
      33.4-5 StronglyConnectedComponents
      33.4-6 PartialOrderByOrderingFunction
    33.5 [33X[0;0YEquivalence Relations[133X
      33.5-1 EquivalenceRelationByPartition
      33.5-2 EquivalenceRelationByRelation
      33.5-3 EquivalenceRelationByPairs
      33.5-4 EquivalenceRelationByProperty
    33.6 [33X[0;0YAttributes of and Operations on Equivalence Relations[133X
      33.6-1 EquivalenceRelationPartition
      33.6-2 GeneratorsOfEquivalenceRelationPartition
      33.6-3 JoinEquivalenceRelations
    33.7 [33X[0;0YEquivalence Classes[133X
      33.7-1 IsEquivalenceClass
      33.7-2 EquivalenceClassRelation
      33.7-3 EquivalenceClasses
      33.7-4 EquivalenceClassOfElement
  34 [33X[0;0YOrderings[133X
    34.1 [33X[0;0YIsOrdering (Filter)[133X
      34.1-1 IsOrdering
      34.1-2 OrderingsFamily
    34.2 [33X[0;0YBuilding new orderings[133X
      34.2-1 OrderingByLessThanFunctionNC
      34.2-2 OrderingByLessThanOrEqualFunctionNC
    34.3 [33X[0;0YProperties and basic functionality[133X
      34.3-1 IsWellFoundedOrdering
      34.3-2 IsTotalOrdering
      34.3-3 IsIncomparableUnder
      34.3-4 FamilyForOrdering
      34.3-5 LessThanFunction
      34.3-6 LessThanOrEqualFunction
      34.3-7 IsLessThanUnder
      34.3-8 IsLessThanOrEqualUnder
    34.4 [33X[0;0YOrderings on families of associative words[133X
      34.4-1 IsOrderingOnFamilyOfAssocWords
      34.4-2 IsTranslationInvariantOrdering
      34.4-3 IsReductionOrdering
      34.4-4 OrderingOnGenerators
      34.4-5 LexicographicOrdering
      34.4-6 ShortLexOrdering
      34.4-7 IsShortLexOrdering
      34.4-8 WeightLexOrdering
      34.4-9 IsWeightLexOrdering
      34.4-10 WeightOfGenerators
      34.4-11 BasicWreathProductOrdering
      34.4-12 IsBasicWreathProductOrdering
      34.4-13 WreathProductOrdering
      34.4-14 IsWreathProductOrdering
      34.4-15 LevelsOfGenerators
  35 [33X[0;0YMagmas[133X
    35.1 [33X[0;0YMagma Categories[133X
      35.1-1 IsMagma
      35.1-2 IsMagmaWithOne
      35.1-3 IsMagmaWithInversesIfNonzero
      35.1-4 IsMagmaWithInverses
    35.2 [33X[0;0YMagma Generation[133X
      35.2-1 Magma
      35.2-2 MagmaWithOne
      35.2-3 MagmaWithInverses
      35.2-4 MagmaByGenerators
      35.2-5 MagmaWithOneByGenerators
      35.2-6 MagmaWithInversesByGenerators
      35.2-7 Submagma
      35.2-8 SubmagmaWithOne
      35.2-9 SubmagmaWithInverses
      35.2-10 AsMagma
      35.2-11 AsSubmagma
      35.2-12 InjectionZeroMagma
    35.3 [33X[0;0YMagmas Defined by Multiplication Tables[133X
      35.3-1 MagmaByMultiplicationTable
      35.3-2 MagmaWithOneByMultiplicationTable
      35.3-3 MagmaWithInversesByMultiplicationTable
      35.3-4 MagmaElement
      35.3-5 [33X[0;0YMultiplicationTable[133X
    35.4 [33X[0;0YAttributes and Properties for Magmas[133X
      35.4-1 GeneratorsOfMagma
      35.4-2 GeneratorsOfMagmaWithOne
      35.4-3 GeneratorsOfMagmaWithInverses
      35.4-4 Centralizer
      35.4-5 Centre
      35.4-6 Idempotents
      35.4-7 IsAssociative
      35.4-8 IsCentral
      35.4-9 IsCommutative
      35.4-10 MultiplicativeNeutralElement
      35.4-11 MultiplicativeZero
      35.4-12 IsMultiplicativeZero
      35.4-13 SquareRoots
      35.4-14 TrivialSubmagmaWithOne
  36 [33X[0;0YWords[133X
    36.1 [33X[0;0YCategories of Words and Nonassociative Words[133X
      36.1-1 IsWord
      36.1-2 IsWordCollection
      36.1-3 IsNonassocWord
      36.1-4 IsNonassocWordCollection
    36.2 [33X[0;0YComparison of Words[133X
      36.2-1 \=
      36.2-2 \<
    36.3 [33X[0;0YOperations for Words[133X
      36.3-1 MappedWord
    36.4 [33X[0;0YFree Magmas[133X
      36.4-1 [33X[0;0YFreeMagma[133X
      36.4-2 [33X[0;0YFreeMagmaWithOne[133X
    36.5 [33X[0;0YExternal Representation for Nonassociative Words[133X
  37 [33X[0;0YAssociative Words[133X
    37.1 [33X[0;0YCategories of Associative Words[133X
      37.1-1 IsAssocWord
    37.2 [33X[0;0YFree Groups, Monoids and Semigroups[133X
      37.2-1 [33X[0;0YFreeGroup[133X
      37.2-2 IsFreeGroup
      37.2-3 AssignGeneratorVariables
    37.3 [33X[0;0YComparison of Associative Words[133X
      37.3-1 \=
      37.3-2 \<
      37.3-3 IsShortLexLessThanOrEqual
      37.3-4 IsBasicWreathLessThanOrEqual
    37.4 [33X[0;0YOperations for Associative Words[133X
      37.4-1 Length
      37.4-2 ExponentSumWord
      37.4-3 Subword
      37.4-4 PositionWord
      37.4-5 [33X[0;0YSubstitutedWord[133X
      37.4-6 EliminatedWord
    37.5 [33X[0;0YOperations for Associative Words by their Syllables[133X
      37.5-1 NumberSyllables
      37.5-2 ExponentSyllable
      37.5-3 GeneratorSyllable
      37.5-4 SubSyllables
    37.6 [33X[0;0YRepresentations for Associative Words[133X
      37.6-1 IsLetterAssocWordRep
      37.6-2 IsLetterWordsFamily
      37.6-3 IsBLetterAssocWordRep
      37.6-4 IsBLetterWordsFamily
      37.6-5 IsSyllableAssocWordRep
      37.6-6 IsSyllableWordsFamily
      37.6-7 Is16BitsFamily
      37.6-8 LetterRepAssocWord
      37.6-9 AssocWordByLetterRep
    37.7 [33X[0;0YThe External Representation for Associative Words[133X
    37.8 [33X[0;0YStraight Line Programs[133X
      37.8-1 IsStraightLineProgram
      37.8-2 StraightLineProgram
      37.8-3 LinesOfStraightLineProgram
      37.8-4 NrInputsOfStraightLineProgram
      37.8-5 ResultOfStraightLineProgram
      37.8-6 StringOfResultOfStraightLineProgram
      37.8-7 CompositionOfStraightLinePrograms
      37.8-8 IntegratedStraightLineProgram
      37.8-9 RestrictOutputsOfSLP
      37.8-10 IntermediateResultOfSLP
      37.8-11 IntermediateResultOfSLPWithoutOverwrite
      37.8-12 IntermediateResultsOfSLPWithoutOverwrite
      37.8-13 ProductOfStraightLinePrograms
      37.8-14 SlotUsagePattern
    37.9 [33X[0;0YStraight Line Program Elements[133X
      37.9-1 IsStraightLineProgElm
      37.9-2 StraightLineProgElm
      37.9-3 StraightLineProgGens
      37.9-4 EvalStraightLineProgElm
      37.9-5 StretchImportantSLPElement
  38 [33X[0;0YRewriting Systems[133X
    38.1 [33X[0;0YOperations on rewriting systems[133X
      38.1-1 IsRewritingSystem
      38.1-2 Rules
      38.1-3 OrderOfRewritingSystem
      38.1-4 ReducedForm
      38.1-5 [33X[0;0YIsConfluent[133X
      38.1-6 ConfluentRws
      38.1-7 IsReduced
      38.1-8 ReduceRules
      38.1-9 AddRule
      38.1-10 AddRuleReduced
      38.1-11 MakeConfluent
      38.1-12 GeneratorsOfRws
    38.2 [33X[0;0YOperations on elements of the algebra[133X
      38.2-1 ReducedProduct
    38.3 [33X[0;0YProperties of rewriting systems[133X
      38.3-1 IsBuiltFromAdditiveMagmaWithInverses
    38.4 [33X[0;0YRewriting in Groups and Monoids[133X
    38.5 [33X[0;0YDeveloping rewriting systems[133X
  39 [33X[0;0YGroups[133X
    39.1 [33X[0;0YGroup Elements[133X
    39.2 [33X[0;0YCreating Groups[133X
      39.2-1 Group
      39.2-2 GroupByGenerators
      39.2-3 GroupWithGenerators
      39.2-4 GeneratorsOfGroup
      39.2-5 AsGroup
      39.2-6 ConjugateGroup
      39.2-7 IsGroup
      39.2-8 InfoGroup
    39.3 [33X[0;0YSubgroups[133X
      39.3-1 Subgroup
      39.3-2 [33X[0;0YIndex ([5XGAP[105X operation)[133X
      39.3-3 IndexInWholeGroup
      39.3-4 AsSubgroup
      39.3-5 IsSubgroup
      39.3-6 IsNormal
      39.3-7 IsCharacteristicSubgroup
      39.3-8 ConjugateSubgroup
      39.3-9 ConjugateSubgroups
      39.3-10 IsSubnormal
      39.3-11 SubgroupByProperty
      39.3-12 SubgroupShell
    39.4 [33X[0;0YClosures of (Sub)groups[133X
      39.4-1 ClosureGroup
      39.4-2 ClosureGroupAddElm
      39.4-3 ClosureGroupDefault
      39.4-4 ClosureSubgroup
    39.5 [33X[0;0YExpressing Group Elements as Words in Generators[133X
      39.5-1 EpimorphismFromFreeGroup
      39.5-2 Factorization
    39.6 [33X[0;0YStructure Descriptions[133X
      39.6-1 StructureDescription
    39.7 [33X[0;0YCosets[133X
      39.7-1 RightCoset
      39.7-2 RightCosets
      39.7-3 CanonicalRightCosetElement
      39.7-4 IsRightCoset
      39.7-5 CosetDecomposition
    39.8 [33X[0;0YTransversals[133X
      39.8-1 RightTransversal
    39.9 [33X[0;0YDouble Cosets[133X
      39.9-1 DoubleCoset
      39.9-2 RepresentativesContainedRightCosets
      39.9-3 DoubleCosets
      39.9-4 IsDoubleCoset
      39.9-5 DoubleCosetRepsAndSizes
      39.9-6 InfoCoset
    39.10 [33X[0;0YConjugacy Classes[133X
      39.10-1 ConjugacyClass
      39.10-2 ConjugacyClasses
      39.10-3 ConjugacyClassesByRandomSearch
      39.10-4 ConjugacyClassesByOrbits
      39.10-5 NrConjugacyClasses
      39.10-6 RationalClass
      39.10-7 RationalClasses
      39.10-8 GaloisGroup
      39.10-9 [33X[0;0YIsConjugate[133X
      39.10-10 NthRootsInGroup
    39.11 [33X[0;0YNormal Structure[133X
      39.11-1 [33X[0;0YNormalizer[133X
      39.11-2 Core
      39.11-3 PCore
      39.11-4 NormalClosure
      39.11-5 NormalIntersection
      39.11-6 ComplementClassesRepresentatives
      39.11-7 InfoComplement
    39.12 [33X[0;0YSpecific and Parametrized Subgroups[133X
      39.12-1 TrivialSubgroup
      39.12-2 CommutatorSubgroup
      39.12-3 DerivedSubgroup
      39.12-4 CommutatorLength
      39.12-5 FittingSubgroup
      39.12-6 FrattiniSubgroup
      39.12-7 PrefrattiniSubgroup
      39.12-8 PerfectResiduum
      39.12-9 RadicalGroup
      39.12-10 Socle
      39.12-11 SupersolvableResiduum
      39.12-12 PRump
    39.13 [33X[0;0YSylow Subgroups and Hall Subgroups[133X
      39.13-1 SylowSubgroup
      39.13-2 SylowComplement
      39.13-3 HallSubgroup
      39.13-4 SylowSystem
      39.13-5 ComplementSystem
      39.13-6 HallSystem
    39.14 [33X[0;0YSubgroups characterized by prime powers[133X
      39.14-1 Omega
      39.14-2 Agemo
    39.15 [33X[0;0YGroup Properties[133X
      39.15-1 IsCyclic
      39.15-2 IsElementaryAbelian
      39.15-3 IsNilpotentGroup
      39.15-4 NilpotencyClassOfGroup
      39.15-5 IsPerfectGroup
      39.15-6 IsSolvableGroup
      39.15-7 IsPolycyclicGroup
      39.15-8 IsSupersolvableGroup
      39.15-9 IsMonomialGroup
      39.15-10 IsSimpleGroup
      39.15-11 IsAlmostSimpleGroup
      39.15-12 [33X[0;0YIsomorphismTypeInfoFiniteSimpleGroup[133X
      39.15-13 SimpleGroup
      39.15-14 SimpleGroupsIterator
      39.15-15 SmallSimpleGroup
      39.15-16 AllSmallNonabelianSimpleGroups
      39.15-17 IsFinitelyGeneratedGroup
      39.15-18 IsSubsetLocallyFiniteGroup
      39.15-19 IsPGroup
      39.15-20 PrimePGroup
      39.15-21 PClassPGroup
      39.15-22 RankPGroup
      39.15-23 IsPSolvable
      39.15-24 IsPNilpotent
    39.16 [33X[0;0YNumerical Group Attributes[133X
      39.16-1 AbelianInvariants
      39.16-2 Exponent
      39.16-3 EulerianFunction
    39.17 [33X[0;0YSubgroup Series[133X
      39.17-1 ChiefSeries
      39.17-2 ChiefSeriesThrough
      39.17-3 ChiefSeriesUnderAction
      39.17-4 SubnormalSeries
      39.17-5 CompositionSeries
      39.17-6 DisplayCompositionSeries
      39.17-7 DerivedSeriesOfGroup
      39.17-8 DerivedLength
      39.17-9 [33X[0;0YElementaryAbelianSeries[133X
      39.17-10 InvariantElementaryAbelianSeries
      39.17-11 LowerCentralSeriesOfGroup
      39.17-12 UpperCentralSeriesOfGroup
      39.17-13 PCentralSeries
      39.17-14 JenningsSeries
      39.17-15 DimensionsLoewyFactors
      39.17-16 AscendingChain
      39.17-17 IntermediateGroup
      39.17-18 IntermediateSubgroups
    39.18 [33X[0;0YFactor Groups[133X
      39.18-1 NaturalHomomorphismByNormalSubgroup
      39.18-2 FactorGroup
      39.18-3 CommutatorFactorGroup
      39.18-4 MaximalAbelianQuotient
      39.18-5 HasAbelianFactorGroup
      39.18-6 HasElementaryAbelianFactorGroup
      39.18-7 CentralizerModulo
    39.19 [33X[0;0YSets of Subgroups[133X
      39.19-1 ConjugacyClassSubgroups
      39.19-2 IsConjugacyClassSubgroupsRep
      39.19-3 ConjugacyClassesSubgroups
      39.19-4 ConjugacyClassesMaximalSubgroups
      39.19-5 AllSubgroups
      39.19-6 MaximalSubgroupClassReps
      39.19-7 MaximalSubgroups
      39.19-8 NormalSubgroups
      39.19-9 MaximalNormalSubgroups
      39.19-10 MinimalNormalSubgroups
    39.20 [33X[0;0YSubgroup Lattice[133X
      39.20-1 LatticeSubgroups
      39.20-2 ClassElementLattice
      39.20-3 DotFileLatticeSubgroups
      39.20-4 MaximalSubgroupsLattice
      39.20-5 MinimalSupergroupsLattice
      39.20-6 RepresentativesPerfectSubgroups
      39.20-7 ConjugacyClassesPerfectSubgroups
      39.20-8 Zuppos
      39.20-9 InfoLattice
    39.21 [33X[0;0YSpecific Methods for Subgroup Lattice Computations[133X
      39.21-1 LatticeByCyclicExtension
      39.21-2 InvariantSubgroupsElementaryAbelianGroup
      39.21-3 SubgroupsSolvableGroup
      39.21-4 SizeConsiderFunction
      39.21-5 ExactSizeConsiderFunction
      39.21-6 InfoPcSubgroup
    39.22 [33X[0;0YSpecial Generating Sets[133X
      39.22-1 GeneratorsSmallest
      39.22-2 LargestElementGroup
      39.22-3 MinimalGeneratingSet
      39.22-4 SmallGeneratingSet
      39.22-5 IndependentGeneratorsOfAbelianGroup
      39.22-6 IndependentGeneratorExponents
    39.23 [33X[0;0Y1-Cohomology[133X
      39.23-1 [33X[0;0YOneCocycles[133X
      39.23-2 OneCoboundaries
      39.23-3 OCOneCocycles
      39.23-4 ComplementClassesRepresentativesEA
      39.23-5 InfoCoh
    39.24 [33X[0;0YSchur Covers and Multipliers[133X
      39.24-1 EpimorphismSchurCover
      39.24-2 SchurCover
      39.24-3 AbelianInvariantsMultiplier
      39.24-4 Epicentre
      39.24-5 NonabelianExteriorSquare
      39.24-6 EpimorphismNonabelianExteriorSquare
      39.24-7 IsCentralFactor
      39.24-8 [33X[0;0YCovering groups of symmetric groups[133X
      39.24-9 BasicSpinRepresentationOfSymmetricGroup
      39.24-10 SchurCoverOfSymmetricGroup
      39.24-11 DoubleCoverOfAlternatingGroup
    39.25 [33X[0;0YTests for the Availability of Methods[133X
      39.25-1 CanEasilyTestMembership
      39.25-2 CanEasilyComputeWithIndependentGensAbelianGroup
      39.25-3 CanComputeSize
      39.25-4 CanComputeSizeAnySubgroup
      39.25-5 CanComputeIndex
      39.25-6 CanComputeIsSubset
      39.25-7 KnowsHowToDecompose
  40 [33X[0;0YGroup Homomorphisms[133X
    40.1 [33X[0;0YCreating Group Homomorphisms[133X
      40.1-1 GroupHomomorphismByImages
      40.1-2 GroupHomomorphismByImagesNC
      40.1-3 GroupGeneralMappingByImages
      40.1-4 [33X[0;0YGroupHomomorphismByFunction[133X
      40.1-5 AsGroupGeneralMappingByImages
    40.2 [33X[0;0YOperations for Group Homomorphisms[133X
    40.3 [33X[0;0YEfficiency of Homomorphisms[133X
      40.3-1 [33X[0;0YMappings given on generators[133X
      40.3-2 [33X[0;0YAction homomorphisms[133X
      40.3-3 [33X[0;0YMappings given by functions[133X
      40.3-4 [33X[0;0YOther operations[133X
      40.3-5 ImagesSmallestGenerators
    40.4 [33X[0;0YHomomorphism for very large groups[133X
    40.5 [33X[0;0YNice Monomorphisms[133X
      40.5-1 IsHandledByNiceMonomorphism
      40.5-2 NiceMonomorphism
      40.5-3 NiceObject
      40.5-4 IsCanonicalNiceMonomorphism
    40.6 [33X[0;0YGroup Automorphisms[133X
      40.6-1 ConjugatorIsomorphism
      40.6-2 ConjugatorAutomorphism
      40.6-3 InnerAutomorphism
      40.6-4 IsConjugatorIsomorphism
      40.6-5 ConjugatorOfConjugatorIsomorphism
    40.7 [33X[0;0YGroups of Automorphisms[133X
      40.7-1 AutomorphismGroup
      40.7-2 IsGroupOfAutomorphisms
      40.7-3 AutomorphismDomain
      40.7-4 IsAutomorphismGroup
      40.7-5 InnerAutomorphismsAutomorphismGroup
      40.7-6 InducedAutomorphism
    40.8 [33X[0;0YCalculating with Group Automorphisms[133X
      40.8-1 AssignNiceMonomorphismAutomorphismGroup
      40.8-2 NiceMonomorphismAutomGroup
    40.9 [33X[0;0YSearching for Homomorphisms[133X
      40.9-1 IsomorphismGroups
      40.9-2 AllHomomorphismClasses
      40.9-3 AllHomomorphisms
      40.9-4 GQuotients
      40.9-5 IsomorphicSubgroups
      40.9-6 MorClassLoop
    40.10 [33X[0;0YRepresentations for Group Homomorphisms[133X
      40.10-1 IsGroupGeneralMappingByImages
      40.10-2 MappingGeneratorsImages
      40.10-3 IsGroupGeneralMappingByAsGroupGeneralMappingByImages
      40.10-4 IsPreimagesByAsGroupGeneralMappingByImages
      40.10-5 IsPermGroupGeneralMapping
      40.10-6 IsToPermGroupGeneralMappingByImages
      40.10-7 IsGroupGeneralMappingByPcgs
      40.10-8 IsPcGroupGeneralMappingByImages
      40.10-9 IsToPcGroupGeneralMappingByImages
      40.10-10 IsFromFpGroupGeneralMappingByImages
      40.10-11 IsFromFpGroupStdGensGeneralMappingByImages
  41 [33X[0;0YGroup Actions[133X
    41.1 [33X[0;0YAbout Group Actions[133X
    41.2 [33X[0;0YBasic Actions[133X
      41.2-1 OnPoints
      41.2-2 OnRight
      41.2-3 OnLeftInverse
      41.2-4 OnSets
      41.2-5 OnTuples
      41.2-6 OnPairs
      41.2-7 OnSetsSets
      41.2-8 OnSetsDisjointSets
      41.2-9 OnSetsTuples
      41.2-10 OnTuplesSets
      41.2-11 OnTuplesTuples
      41.2-12 OnLines
      41.2-13 OnIndeterminates
      41.2-14 Permuted
      41.2-15 OnSubspacesByCanonicalBasis
    41.3 [33X[0;0YAction on canonical representatives[133X
    41.4 [33X[0;0YOrbits[133X
      41.4-1 Orbit
      41.4-2 Orbits
      41.4-3 [33X[0;0YOrbitsDomain[133X
      41.4-4 OrbitLength
      41.4-5 [33X[0;0YOrbitLengths[133X
      41.4-6 [33X[0;0YOrbitLengthsDomain[133X
    41.5 [33X[0;0YStabilizers[133X
      41.5-1 OrbitStabilizer
      41.5-2 Stabilizer
      41.5-3 OrbitStabilizerAlgorithm
    41.6 [33X[0;0YElements with Prescribed Images[133X
      41.6-1 RepresentativeAction
    41.7 [33X[0;0YThe Permutation Image of an Action[133X
      41.7-1 [33X[0;0YActionHomomorphism[133X
      41.7-2 Action
      41.7-3 SparseActionHomomorphism
    41.8 [33X[0;0YAction of a group on itself[133X
      41.8-1 FactorCosetAction
      41.8-2 RegularActionHomomorphism
      41.8-3 AbelianSubfactorAction
    41.9 [33X[0;0YPermutations Induced by Elements and Cycles[133X
      41.9-1 [33X[0;0YPermutation[133X
      41.9-2 PermutationCycle
      41.9-3 Cycle
      41.9-4 CycleLength
      41.9-5 Cycles
      41.9-6 CycleLengths
      41.9-7 [33X[0;0YCycleIndex[133X
    41.10 [33X[0;0YTests for Actions[133X
      41.10-1 [33X[0;0YIsTransitive[133X
      41.10-2 [33X[0;0YTransitivity[133X
      41.10-3 [33X[0;0YRankAction[133X
      41.10-4 [33X[0;0YIsSemiRegular[133X
      41.10-5 [33X[0;0YIsRegular[133X
      41.10-6 [33X[0;0YEarns[133X
      41.10-7 [33X[0;0YIsPrimitive[133X
    41.11 [33X[0;0YBlock Systems[133X
      41.11-1 [33X[0;0YBlocks[133X
      41.11-2 [33X[0;0YMaximalBlocks[133X
      41.11-3 [33X[0;0YRepresentativesMinimalBlocks[133X
      41.11-4 AllBlocks
    41.12 [33X[0;0YExternal Sets[133X
      41.12-1 IsExternalSet
      41.12-2 ExternalSet
      41.12-3 ActingDomain
      41.12-4 FunctionAction
      41.12-5 HomeEnumerator
      41.12-6 IsExternalSubset
      41.12-7 ExternalSubset
      41.12-8 IsExternalOrbit
      41.12-9 ExternalOrbit
      41.12-10 StabilizerOfExternalSet
      41.12-11 [33X[0;0YExternalOrbits[133X
      41.12-12 [33X[0;0YExternalOrbitsStabilizers[133X
      41.12-13 CanonicalRepresentativeOfExternalSet
      41.12-14 CanonicalRepresentativeDeterminatorOfExternalSet
      41.12-15 ActorOfExternalSet
      41.12-16 UnderlyingExternalSet
      41.12-17 SurjectiveActionHomomorphismAttr
  42 [33X[0;0YPermutations[133X
    42.1 [33X[0;0YIsPerm (Filter)[133X
      42.1-1 IsPerm
      42.1-2 IsPermCollection
      42.1-3 PermutationsFamily
    42.2 [33X[0;0YComparison of Permutations[133X
      42.2-1 \=
      42.2-2 DistancePerms
      42.2-3 SmallestGeneratorPerm
    42.3 [33X[0;0YMoved Points of Permutations[133X
      42.3-1 SmallestMovedPoint
      42.3-2 LargestMovedPoint
      42.3-3 MovedPoints
      42.3-4 NrMovedPoints
    42.4 [33X[0;0YSign and Cycle Structure[133X
      42.4-1 SignPerm
      42.4-2 CycleStructurePerm
    42.5 [33X[0;0YCreating Permutations[133X
      42.5-1 ListPerm
      42.5-2 PermList
      42.5-3 MappingPermListList
      42.5-4 RestrictedPerm
  43 [33X[0;0YPermutation Groups[133X
    43.1 [33X[0;0YIsPermGroup (Filter)[133X
      43.1-1 IsPermGroup
    43.2 [33X[0;0YThe Natural Action[133X
      43.2-1 OrbitPerms
      43.2-2 OrbitsPerms
    43.3 [33X[0;0YComputing a Permutation Representation[133X
      43.3-1 IsomorphismPermGroup
      43.3-2 SmallerDegreePermutationRepresentation
    43.4 [33X[0;0YSymmetric and Alternating Groups[133X
      43.4-1 IsNaturalSymmetricGroup
      43.4-2 IsSymmetricGroup
      43.4-3 IsAlternatingGroup
      43.4-4 SymmetricParentGroup
    43.5 [33X[0;0YPrimitive Groups[133X
      43.5-1 ONanScottType
      43.5-2 SocleTypePrimitiveGroup
    43.6 [33X[0;0YStabilizer Chains[133X
    43.7 [33X[0;0YRandomized Methods for Permutation Groups[133X
    43.8 [33X[0;0YConstruction of Stabilizer Chains[133X
      43.8-1 StabChain
      43.8-2 StabChainOptions
      43.8-3 DefaultStabChainOptions
      43.8-4 StabChainBaseStrongGenerators
      43.8-5 MinimalStabChain
    43.9 [33X[0;0YStabilizer Chain Records[133X
    43.10 [33X[0;0YOperations for Stabilizer Chains[133X
      43.10-1 BaseStabChain
      43.10-2 BaseOfGroup
      43.10-3 SizeStabChain
      43.10-4 StrongGeneratorsStabChain
      43.10-5 GroupStabChain
      43.10-6 OrbitStabChain
      43.10-7 IndicesStabChain
      43.10-8 ListStabChain
      43.10-9 ElementsStabChain
      43.10-10 InverseRepresentative
      43.10-11 SiftedPermutation
      43.10-12 MinimalElementCosetStabChain
      43.10-13 LargestElementStabChain
      43.10-14 ApproximateSuborbitsStabilizerPermGroup
    43.11 [33X[0;0YLow Level Routines to Modify and Create Stabilizer Chains[133X
      43.11-1 CopyStabChain
      43.11-2 CopyOptionsDefaults
      43.11-3 ChangeStabChain
      43.11-4 ExtendStabChain
      43.11-5 ReduceStabChain
      43.11-6 RemoveStabChain
      43.11-7 EmptyStabChain
      43.11-8 InsertTrivialStabilizer
      43.11-9 IsFixedStabilizer
      43.11-10 AddGeneratorsExtendSchreierTree
    43.12 [33X[0;0YBacktrack[133X
      43.12-1 SubgroupProperty
      43.12-2 ElementProperty
      43.12-3 TwoClosure
      43.12-4 InfoBckt
    43.13 [33X[0;0YWorking with large degree permutation groups[133X
  44 [33X[0;0YMatrix Groups[133X
    44.1 [33X[0;0YIsMatrixGroup (Filter)[133X
      44.1-1 IsMatrixGroup
    44.2 [33X[0;0YAttributes and Properties for Matrix Groups[133X
      44.2-1 DimensionOfMatrixGroup
      44.2-2 DefaultFieldOfMatrixGroup
      44.2-3 FieldOfMatrixGroup
      44.2-4 TransposedMatrixGroup
      44.2-5 IsFFEMatrixGroup
    44.3 [33X[0;0YActions of Matrix Groups[133X
      44.3-1 ProjectiveActionOnFullSpace
      44.3-2 ProjectiveActionHomomorphismMatrixGroup
      44.3-3 BlowUpIsomorphism
    44.4 [33X[0;0YGL and SL[133X
      44.4-1 IsGeneralLinearGroup
      44.4-2 IsNaturalGL
      44.4-3 IsSpecialLinearGroup
      44.4-4 IsNaturalSL
      44.4-5 IsSubgroupSL
    44.5 [33X[0;0YInvariant Forms[133X
      44.5-1 InvariantBilinearForm
      44.5-2 IsFullSubgroupGLorSLRespectingBilinearForm
      44.5-3 InvariantSesquilinearForm
      44.5-4 IsFullSubgroupGLorSLRespectingSesquilinearForm
      44.5-5 InvariantQuadraticForm
      44.5-6 IsFullSubgroupGLorSLRespectingQuadraticForm
    44.6 [33X[0;0YMatrix Groups in Characteristic 0[133X
      44.6-1 IsCyclotomicMatrixGroup
      44.6-2 IsRationalMatrixGroup
      44.6-3 IsIntegerMatrixGroup
      44.6-4 IsNaturalGLnZ
      44.6-5 IsNaturalSLnZ
      44.6-6 InvariantLattice
      44.6-7 NormalizerInGLnZ
      44.6-8 CentralizerInGLnZ
      44.6-9 ZClassRepsQClass
      44.6-10 IsBravaisGroup
      44.6-11 BravaisGroup
      44.6-12 BravaisSubgroups
      44.6-13 BravaisSupergroups
      44.6-14 NormalizerInGLnZBravaisGroup
    44.7 [33X[0;0YActing OnRight and OnLeft[133X
      44.7-1 CrystGroupDefaultAction
      44.7-2 SetCrystGroupDefaultAction
  45 [33X[0;0YPolycyclic Groups[133X
    45.1 [33X[0;0YPolycyclic Generating Systems[133X
    45.2 [33X[0;0YComputing a Pcgs[133X
      45.2-1 Pcgs
      45.2-2 IsPcgs
      45.2-3 CanEasilyComputePcgs
    45.3 [33X[0;0YDefining a Pcgs Yourself[133X
      45.3-1 PcgsByPcSequence
    45.4 [33X[0;0YElementary Operations for a Pcgs[133X
      45.4-1 RelativeOrders
      45.4-2 IsFiniteOrdersPcgs
      45.4-3 IsPrimeOrdersPcgs
      45.4-4 PcSeries
      45.4-5 GroupOfPcgs
      45.4-6 OneOfPcgs
    45.5 [33X[0;0YElementary Operations for a Pcgs and an Element[133X
      45.5-1 RelativeOrderOfPcElement
      45.5-2 ExponentOfPcElement
      45.5-3 ExponentsOfPcElement
      45.5-4 DepthOfPcElement
      45.5-5 LeadingExponentOfPcElement
      45.5-6 PcElementByExponents
      45.5-7 LinearCombinationPcgs
      45.5-8 SiftedPcElement
      45.5-9 CanonicalPcElement
      45.5-10 ReducedPcElement
      45.5-11 CleanedTailPcElement
      45.5-12 HeadPcElementByNumber
    45.6 [33X[0;0YExponents of Special Products[133X
      45.6-1 ExponentsConjugateLayer
      45.6-2 ExponentsOfRelativePower
      45.6-3 ExponentsOfConjugate
      45.6-4 ExponentsOfCommutator
    45.7 [33X[0;0YSubgroups of Polycyclic Groups - Induced Pcgs[133X
      45.7-1 IsInducedPcgs
      45.7-2 InducedPcgsByPcSequence
      45.7-3 ParentPcgs
      45.7-4 InducedPcgs
      45.7-5 InducedPcgsByGenerators
      45.7-6 InducedPcgsByPcSequenceAndGenerators
      45.7-7 LeadCoeffsIGS
      45.7-8 ExtendedPcgs
      45.7-9 SubgroupByPcgs
    45.8 [33X[0;0YSubgroups of Polycyclic Groups – Canonical Pcgs[133X
      45.8-1 IsCanonicalPcgs
      45.8-2 CanonicalPcgs
    45.9 [33X[0;0YFactor Groups of Polycyclic Groups – Modulo Pcgs[133X
      45.9-1 ModuloPcgs
      45.9-2 IsModuloPcgs
      45.9-3 NumeratorOfModuloPcgs
      45.9-4 DenominatorOfModuloPcgs
      45.9-5 \mod
      45.9-6 CorrespondingGeneratorsByModuloPcgs
      45.9-7 CanonicalPcgsByGeneratorsWithImages
    45.10 [33X[0;0YFactor Groups of Polycyclic Groups in their Own Representation[133X
      45.10-1 ProjectedPcElement
      45.10-2 ProjectedInducedPcgs
      45.10-3 LiftedPcElement
      45.10-4 LiftedInducedPcgs
    45.11 [33X[0;0YPcgs and Normal Series[133X
      45.11-1 IsPcgsElementaryAbelianSeries
      45.11-2 PcgsElementaryAbelianSeries
      45.11-3 IndicesEANormalSteps
      45.11-4 EANormalSeriesByPcgs
      45.11-5 IsPcgsCentralSeries
      45.11-6 PcgsCentralSeries
      45.11-7 IndicesCentralNormalSteps
      45.11-8 CentralNormalSeriesByPcgs
      45.11-9 IsPcgsPCentralSeriesPGroup
      45.11-10 PcgsPCentralSeriesPGroup
      45.11-11 IndicesPCentralNormalStepsPGroup
      45.11-12 PCentralNormalSeriesByPcgsPGroup
      45.11-13 IsPcgsChiefSeries
      45.11-14 PcgsChiefSeries
      45.11-15 IndicesChiefNormalSteps
      45.11-16 ChiefNormalSeriesByPcgs
      45.11-17 IndicesNormalSteps
      45.11-18 NormalSeriesByPcgs
    45.12 [33X[0;0YSum and Intersection of Pcgs[133X
      45.12-1 SumFactorizationFunctionPcgs
    45.13 [33X[0;0YSpecial Pcgs[133X
      45.13-1 IsSpecialPcgs
      45.13-2 [33X[0;0YSpecialPcgs[133X
      45.13-3 LGWeights
      45.13-4 LGLayers
      45.13-5 LGFirst
      45.13-6 LGLength
      45.13-7 IsInducedPcgsWrtSpecialPcgs
      45.13-8 InducedPcgsWrtSpecialPcgs
    45.14 [33X[0;0YAction on Subfactors Defined by a Pcgs[133X
      45.14-1 VectorSpaceByPcgsOfElementaryAbelianGroup
      45.14-2 LinearAction
      45.14-3 LinearActionLayer
      45.14-4 AffineAction
      45.14-5 AffineActionLayer
    45.15 [33X[0;0YOrbit Stabilizer Methods for Polycyclic Groups[133X
      45.15-1 StabilizerPcgs
      45.15-2 Pcgs_OrbitStabilizer
    45.16 [33X[0;0YOperations which have Special Methods for Groups with Pcgs[133X
    45.17 [33X[0;0YConjugacy Classes in Solvable Groups[133X
      45.17-1 ClassesSolvableGroup
      45.17-2 CentralizerSizeLimitConsiderFunction
  46 [33X[0;0YPc Groups[133X
    46.1 [33X[0;0YThe family pcgs[133X
      46.1-1 FamilyPcgs
      46.1-2 IsFamilyPcgs
      46.1-3 InducedPcgsWrtFamilyPcgs
      46.1-4 IsParentPcgsFamilyPcgs
    46.2 [33X[0;0YElements of pc groups[133X
      46.2-1 [33X[0;0YComparison of elements of pc groups[133X
      46.2-2 [33X[0;0YArithmetic operations for elements of pc groups[133X
    46.3 [33X[0;0YPc groups versus fp groups[133X
      46.3-1 IsPcGroup
      46.3-2 IsomorphismFpGroupByPcgs
    46.4 [33X[0;0YConstructing Pc Groups[133X
      46.4-1 PcGroupFpGroup
      46.4-2 SingleCollector
      46.4-3 SetConjugate
      46.4-4 SetCommutator
      46.4-5 SetPower
      46.4-6 GroupByRws
      46.4-7 IsConfluent
      46.4-8 IsomorphismRefinedPcGroup
      46.4-9 RefinedPcGroup
    46.5 [33X[0;0YComputing Pc Groups[133X
      46.5-1 PcGroupWithPcgs
      46.5-2 IsomorphismPcGroup
      46.5-3 IsomorphismSpecialPcGroup
    46.6 [33X[0;0YSaving a Pc Group[133X
      46.6-1 GapInputPcGroup
    46.7 [33X[0;0YOperations for Pc Groups[133X
    46.8 [33X[0;0Y[22X2[122X-Cohomology and Extensions[133X
      46.8-1 TwoCoboundaries
      46.8-2 TwoCocycles
      46.8-3 TwoCohomology
      46.8-4 Extensions
      46.8-5 Extension
      46.8-6 SplitExtension
      46.8-7 ModuleOfExtension
      46.8-8 CompatiblePairs
      46.8-9 ExtensionRepresentatives
      46.8-10 SplitExtensions
    46.9 [33X[0;0YCoding a Pc Presentation[133X
      46.9-1 CodePcgs
      46.9-2 CodePcGroup
      46.9-3 PcGroupCode
    46.10 [33X[0;0YRandom Isomorphism Testing[133X
      46.10-1 RandomIsomorphismTest
  47 [33X[0;0YFinitely Presented Groups[133X
    47.1 [33X[0;0YIsSubgroupFpGroup and IsFpGroup[133X
      47.1-1 IsSubgroupFpGroup
      47.1-2 IsFpGroup
      47.1-3 InfoFpGroup
    47.2 [33X[0;0YCreating Finitely Presented Groups[133X
      47.2-1 \/
      47.2-2 FactorGroupFpGroupByRels
      47.2-3 ParseRelators
      47.2-4 StringFactorizationWord
    47.3 [33X[0;0YComparison of Elements of Finitely Presented Groups[133X
      47.3-1 \=
      47.3-2 \<
      47.3-3 FpElmComparisonMethod
      47.3-4 SetReducedMultiplication
    47.4 [33X[0;0YPreimages in the Free Group[133X
      47.4-1 FreeGroupOfFpGroup
      47.4-2 FreeGeneratorsOfFpGroup
      47.4-3 RelatorsOfFpGroup
      47.4-4 UnderlyingElement
      47.4-5 ElementOfFpGroup
    47.5 [33X[0;0YOperations for Finitely Presented Groups[133X
      47.5-1 PseudoRandom
    47.6 [33X[0;0YCoset Tables and Coset Enumeration[133X
      47.6-1 CosetTable
      47.6-2 TracedCosetFpGroup
      47.6-3 FactorCosetAction
      47.6-4 CosetTableBySubgroup
      47.6-5 CosetTableFromGensAndRels
      47.6-6 CosetTableDefaultMaxLimit
      47.6-7 CosetTableDefaultLimit
      47.6-8 MostFrequentGeneratorFpGroup
      47.6-9 IndicesInvolutaryGenerators
    47.7 [33X[0;0YStandardization of coset tables[133X
      47.7-1 CosetTableStandard
      47.7-2 StandardizeTable
    47.8 [33X[0;0YCoset tables for subgroups in the whole group[133X
      47.8-1 CosetTableInWholeGroup
      47.8-2 SubgroupOfWholeGroupByCosetTable
    47.9 [33X[0;0YAugmented Coset Tables and Rewriting[133X
      47.9-1 AugmentedCosetTableInWholeGroup
      47.9-2 AugmentedCosetTableMtc
      47.9-3 AugmentedCosetTableRrs
      47.9-4 RewriteWord
    47.10 [33X[0;0YLow Index Subgroups[133X
      47.10-1 LowIndexSubgroupsFpGroupIterator
    47.11 [33X[0;0YConverting Groups to Finitely Presented Groups[133X
      47.11-1 IsomorphismFpGroup
      47.11-2 IsomorphismFpGroupByGenerators
    47.12 [33X[0;0YNew Presentations and Presentations for Subgroups[133X
      47.12-1 IsomorphismSimplifiedFpGroup
    47.13 [33X[0;0YPreimages under Homomorphisms from an FpGroup[133X
      47.13-1 SubgroupOfWholeGroupByQuotientSubgroup
      47.13-2 IsSubgroupOfWholeGroupByQuotientRep
      47.13-3 AsSubgroupOfWholeGroupByQuotient
      47.13-4 DefiningQuotientHomomorphism
    47.14 [33X[0;0YQuotient Methods[133X
      47.14-1 PQuotient
      47.14-2 EpimorphismQuotientSystem
      47.14-3 EpimorphismPGroup
      47.14-4 EpimorphismNilpotentQuotient
      47.14-5 SolvableQuotient
      47.14-6 EpimorphismSolvableQuotient
      47.14-7 LargerQuotientBySubgroupAbelianization
    47.15 [33X[0;0YAbelian Invariants for Subgroups[133X
      47.15-1 AbelianInvariantsSubgroupFpGroup
      47.15-2 AbelianInvariantsSubgroupFpGroupMtc
      47.15-3 [33X[0;0YAbelianInvariantsSubgroupFpGroupRrs[133X
      47.15-4 AbelianInvariantsNormalClosureFpGroup
      47.15-5 AbelianInvariantsNormalClosureFpGroupRrs
    47.16 [33X[0;0YTesting Finiteness of Finitely Presented Groups[133X
      47.16-1 IsInfiniteAbelianizationGroup
      47.16-2 NewmanInfinityCriterion
  48 [33X[0;0YPresentations and Tietze Transformations[133X
    48.1 [33X[0;0YCreating Presentations[133X
      48.1-1 PresentationFpGroup
      48.1-2 TzSort
      48.1-3 GeneratorsOfPresentation
      48.1-4 FpGroupPresentation
      48.1-5 PresentationViaCosetTable
      48.1-6 SimplifiedFpGroup
    48.2 [33X[0;0YSubgroup Presentations[133X
      48.2-1 PresentationSubgroup
      48.2-2 [33X[0;0YPresentationSubgroupRrs[133X
      48.2-3 PrimaryGeneratorWords
      48.2-4 PresentationSubgroupMtc
      48.2-5 PresentationNormalClosureRrs
      48.2-6 PresentationNormalClosure
    48.3 [33X[0;0YRelators in a Presentation[133X
      48.3-1 TietzeWordAbstractWord
      48.3-2 AbstractWordTietzeWord
    48.4 [33X[0;0YPrinting Presentations[133X
      48.4-1 TzPrintGenerators
      48.4-2 TzPrintRelators
      48.4-3 TzPrintLengths
      48.4-4 TzPrintStatus
      48.4-5 TzPrintPresentation
      48.4-6 TzPrint
      48.4-7 TzPrintPairs
    48.5 [33X[0;0YChanging Presentations[133X
      48.5-1 AddGenerator
      48.5-2 TzNewGenerator
      48.5-3 AddRelator
      48.5-4 RemoveRelator
    48.6 [33X[0;0YTietze Transformations[133X
      48.6-1 TzGo
      48.6-2 SimplifyPresentation
      48.6-3 TzGoGo
    48.7 [33X[0;0YElementary Tietze Transformations[133X
      48.7-1 [33X[0;0YTzEliminate[133X
      48.7-2 TzSearch
      48.7-3 TzSearchEqual
      48.7-4 TzFindCyclicJoins
    48.8 [33X[0;0YTietze Transformations that introduce new Generators[133X
      48.8-1 [33X[0;0YTzSubstitute[133X
      48.8-2 TzSubstituteCyclicJoins
    48.9 [33X[0;0YTracing generator images through Tietze transformations[133X
      48.9-1 TzInitGeneratorImages
      48.9-2 OldGeneratorsOfPresentation
      48.9-3 TzImagesOldGens
      48.9-4 TzPreImagesNewGens
      48.9-5 TzPrintGeneratorImages
    48.10 [33X[0;0YThe Decoding Tree Procedure[133X
      48.10-1 DecodeTree
    48.11 [33X[0;0YTietze Options[133X
      48.11-1 TzOptions
      48.11-2 TzPrintOptions
  49 [33X[0;0YGroup Products[133X
    49.1 [33X[0;0YDirect Products[133X
      49.1-1 DirectProduct
    49.2 [33X[0;0YSemidirect Products[133X
      49.2-1 [33X[0;0YSemidirectProduct[133X
    49.3 [33X[0;0YSubdirect Products[133X
      49.3-1 SubdirectProduct
      49.3-2 SubdirectProducts
    49.4 [33X[0;0YWreath Products[133X
      49.4-1 WreathProduct
      49.4-2 WreathProductImprimitiveAction
      49.4-3 WreathProductProductAction
      49.4-4 KuKGenerators
    49.5 [33X[0;0YFree Products[133X
      49.5-1 [33X[0;0YFreeProduct[133X
    49.6 [33X[0;0YEmbeddings and Projections for Group Products[133X
      49.6-1 Embedding
      49.6-2 Projection
  50 [33X[0;0YGroup Libraries[133X
    50.1 [33X[0;0YBasic Groups[133X
      50.1-1 TrivialGroup
      50.1-2 CyclicGroup
      50.1-3 AbelianGroup
      50.1-4 ElementaryAbelianGroup
      50.1-5 DihedralGroup
      50.1-6 QuaternionGroup
      50.1-7 ExtraspecialGroup
      50.1-8 [33X[0;0YAlternatingGroup[133X
      50.1-9 [33X[0;0YSymmetricGroup[133X
      50.1-10 MathieuGroup
      50.1-11 SuzukiGroup
      50.1-12 ReeGroup
    50.2 [33X[0;0YClassical Groups[133X
      50.2-1 [33X[0;0YGeneralLinearGroup[133X
      50.2-2 [33X[0;0YSpecialLinearGroup[133X
      50.2-3 GeneralUnitaryGroup
      50.2-4 SpecialUnitaryGroup
      50.2-5 [33X[0;0YSymplecticGroup[133X
      50.2-6 GeneralOrthogonalGroup
      50.2-7 SpecialOrthogonalGroup
      50.2-8 Omega
      50.2-9 GeneralSemilinearGroup
      50.2-10 SpecialSemilinearGroup
      50.2-11 ProjectiveGeneralLinearGroup
      50.2-12 ProjectiveSpecialLinearGroup
      50.2-13 ProjectiveGeneralUnitaryGroup
      50.2-14 ProjectiveSpecialUnitaryGroup
      50.2-15 ProjectiveSymplecticGroup
      50.2-16 ProjectiveOmega
    50.3 [33X[0;0YConjugacy Classes in Classical Groups[133X
      50.3-1 NrConjugacyClassesGL
    50.4 [33X[0;0YConstructors for Basic Groups[133X
    50.5 [33X[0;0YSelection Functions[133X
    50.6 [33X[0;0YTransitive Permutation Groups[133X
      50.6-1 TransitiveGroup
      50.6-2 NrTransitiveGroups
      50.6-3 TransitiveIdentification
    50.7 [33X[0;0YSmall Groups[133X
      50.7-1 SmallGroup
      50.7-2 AllSmallGroups
      50.7-3 OneSmallGroup
      50.7-4 NumberSmallGroups
      50.7-5 IdSmallGroup
      50.7-6 IdsOfAllSmallGroups
      50.7-7 IdGap3SolvableGroup
      50.7-8 SmallGroupsInformation
      50.7-9 UnloadSmallGroupsData
    50.8 [33X[0;0YFinite Perfect Groups[133X
      50.8-1 SizesPerfectGroups
      50.8-2 [33X[0;0YPerfectGroup[133X
      50.8-3 PerfectIdentification
      50.8-4 NumberPerfectGroups
      50.8-5 NumberPerfectLibraryGroups
      50.8-6 SizeNumbersPerfectGroups
      50.8-7 [33X[0;0YDisplayInformationPerfectGroups[133X
      50.8-8 [33X[0;0YMore about the Perfect Groups Library[133X
    50.9 [33X[0;0YPrimitive Permutation Groups[133X
      50.9-1 PrimitiveGroup
      50.9-2 NrPrimitiveGroups
      50.9-3 PrimitiveGroupsIterator
      50.9-4 COHORTS_PRIMITIVE_GROUPS
    50.10 [33X[0;0YIndex numbers of primitive groups[133X
      50.10-1 PrimitiveIdentification
      50.10-2 SimsNo
      50.10-3 PRIMITIVE_INDICES_MAGMA
    50.11 [33X[0;0YIrreducible Solvable Matrix Groups[133X
      50.11-1 IrreducibleSolvableGroupMS
      50.11-2 NumberIrreducibleSolvableGroups
      50.11-3 AllIrreducibleSolvableGroups
      50.11-4 OneIrreducibleSolvableGroup
      50.11-5 PrimitiveIndexIrreducibleSolvableGroup
      50.11-6 IrreducibleSolvableGroup
    50.12 [33X[0;0YIrreducible Maximal Finite Integral Matrix Groups[133X
      50.12-1 ImfNumberQQClasses
      50.12-2 DisplayImfInvariants
      50.12-3 ImfInvariants
      50.12-4 ImfMatrixGroup
      50.12-5 IsomorphismPermGroup
      50.12-6 IsomorphismPermGroupImfGroup
  51 [33X[0;0YSemigroups[133X
    51.1 [33X[0;0YIsSemigroup (Filter)[133X
      51.1-1 IsSemigroup
      51.1-2 [33X[0;0YSemigroup[133X
      51.1-3 Subsemigroup
      51.1-4 SemigroupByGenerators
      51.1-5 AsSemigroup
      51.1-6 AsSubsemigroup
      51.1-7 GeneratorsOfSemigroup
      51.1-8 [33X[0;0YFreeSemigroup[133X
      51.1-9 SemigroupByMultiplicationTable
    51.2 [33X[0;0YProperties of Semigroups[133X
      51.2-1 IsRegularSemigroup
      51.2-2 IsRegularSemigroupElement
      51.2-3 IsSimpleSemigroup
      51.2-4 IsZeroSimpleSemigroup
      51.2-5 IsZeroGroup
      51.2-6 IsReesCongruenceSemigroup
    51.3 [33X[0;0YMaking transformation semigroups[133X
      51.3-1 IsTransformationSemigroup
      51.3-2 DegreeOfTransformationSemigroup
      51.3-3 IsomorphismTransformationSemigroup
      51.3-4 IsFullTransformationSemigroup
      51.3-5 FullTransformationSemigroup
    51.4 [33X[0;0YIdeals of semigroups[133X
      51.4-1 SemigroupIdealByGenerators
      51.4-2 ReesCongruenceOfSemigroupIdeal
      51.4-3 IsLeftSemigroupIdeal
    51.5 [33X[0;0YCongruences for semigroups[133X
      51.5-1 IsSemigroupCongruence
      51.5-2 IsReesCongruence
    51.6 [33X[0;0YQuotients[133X
      51.6-1 IsQuotientSemigroup
      51.6-2 HomomorphismQuotientSemigroup
      51.6-3 QuotientSemigroupPreimage
    51.7 [33X[0;0YGreen's Relations[133X
      51.7-1 GreensRRelation
      51.7-2 IsGreensRelation
      51.7-3 IsGreensClass
      51.7-4 IsGreensLessThanOrEqual
      51.7-5 RClassOfHClass
      51.7-6 EggBoxOfDClass
      51.7-7 DisplayEggBoxOfDClass
      51.7-8 GreensRClassOfElement
      51.7-9 GreensRClasses
      51.7-10 GroupHClassOfGreensDClass
      51.7-11 IsGroupHClass
      51.7-12 IsRegularDClass
    51.8 [33X[0;0YRees Matrix Semigroups[133X
      51.8-1 ReesMatrixSemigroup
      51.8-2 ReesZeroMatrixSemigroup
      51.8-3 IsReesMatrixSemigroup
      51.8-4 IsReesZeroMatrixSemigroup
      51.8-5 ReesMatrixSemigroupElement
      51.8-6 IsReesMatrixSemigroupElement
      51.8-7 SandwichMatrixOfReesMatrixSemigroup
      51.8-8 RowIndexOfReesMatrixSemigroupElement
      51.8-9 ReesZeroMatrixSemigroupElementIsZero
      51.8-10 AssociatedReesMatrixSemigroupOfDClass
      51.8-11 IsomorphismReesMatrixSemigroup
  52 [33X[0;0YMonoids[133X
    52.1 [33X[0;0YFunctions for Monoids[133X
      52.1-1 IsMonoid
      52.1-2 [33X[0;0YMonoid[133X
      52.1-3 Submonoid
      52.1-4 MonoidByGenerators
      52.1-5 AsMonoid
      52.1-6 AsSubmonoid
      52.1-7 GeneratorsOfMonoid
      52.1-8 TrivialSubmonoid
      52.1-9 [33X[0;0YFreeMonoid[133X
      52.1-10 MonoidByMultiplicationTable
  53 [33X[0;0YFinitely Presented Semigroups and Monoids[133X
    53.1 [33X[0;0YIsSubsemigroupFpSemigroup (Filter)[133X
      53.1-1 IsSubsemigroupFpSemigroup
      53.1-2 IsSubmonoidFpMonoid
      53.1-3 IsFpSemigroup
      53.1-4 IsFpMonoid
      53.1-5 IsElementOfFpSemigroup
      53.1-6 IsElementOfFpMonoid
      53.1-7 FpGrpMonSmgOfFpGrpMonSmgElement
    53.2 [33X[0;0YCreating Finitely Presented Semigroups[133X
      53.2-1 \/
      53.2-2 FactorFreeSemigroupByRelations
      53.2-3 IsomorphismFpSemigroup
    53.3 [33X[0;0YComparison of Elements of Finitely Presented Semigroups[133X
      53.3-1 \=
    53.4 [33X[0;0YPreimages in the Free Semigroup[133X
      53.4-1 UnderlyingElement
      53.4-2 ElementOfFpSemigroup
      53.4-3 FreeSemigroupOfFpSemigroup
      53.4-4 FreeGeneratorsOfFpSemigroup
      53.4-5 RelationsOfFpSemigroup
    53.5 [33X[0;0YFinitely presented monoids[133X
      53.5-1 \/
    53.6 [33X[0;0YRewriting Systems and the Knuth-Bendix Procedure[133X
      53.6-1 ReducedConfluentRewritingSystem
      53.6-2 KB_REW
      53.6-3 [33X[0;0YKnuthBendixRewritingSystem[133X
      53.6-4 SemigroupOfRewritingSystem
      53.6-5 MonoidOfRewritingSystem
      53.6-6 FreeSemigroupOfRewritingSystem
      53.6-7 FreeMonoidOfRewritingSystem
    53.7 [33X[0;0YTodd-Coxeter Procedure[133X
      53.7-1 CosetTableOfFpSemigroup
  54 [33X[0;0YTransformations[133X
    54.1 [33X[0;0YFunctions for Transformations[133X
      54.1-1 IsTransformation
      54.1-2 TransformationFamily
      54.1-3 Transformation
      54.1-4 IdentityTransformation
      54.1-5 RandomTransformation
      54.1-6 DegreeOfTransformation
      54.1-7 ImageListOfTransformation
      54.1-8 ImageSetOfTransformation
      54.1-9 RankOfTransformation
      54.1-10 KernelOfTransformation
      54.1-11 PreimagesOfTransformation
      54.1-12 RestrictedTransformation
      54.1-13 AsTransformation
      54.1-14 PermLeftQuoTransformation
      54.1-15 BinaryRelationTransformation
      54.1-16 TransformationRelation
  55 [33X[0;0YAdditive Magmas[133X
    55.1 [33X[0;0Y(Near-)Additive Magma Categories[133X
      55.1-1 IsNearAdditiveMagma
      55.1-2 IsNearAdditiveMagmaWithZero
      55.1-3 IsNearAdditiveGroup
      55.1-4 IsAdditiveMagma
      55.1-5 IsAdditiveMagmaWithZero
      55.1-6 IsAdditiveGroup
    55.2 [33X[0;0Y(Near-)Additive Magma Generation[133X
      55.2-1 NearAdditiveMagma
      55.2-2 NearAdditiveMagmaWithZero
      55.2-3 NearAdditiveGroup
      55.2-4 NearAdditiveMagmaByGenerators
      55.2-5 NearAdditiveMagmaWithZeroByGenerators
      55.2-6 NearAdditiveGroupByGenerators
      55.2-7 SubnearAdditiveMagma
      55.2-8 SubnearAdditiveMagmaWithZero
      55.2-9 SubnearAdditiveGroup
    55.3 [33X[0;0YAttributes and Properties for (Near-)Additive Magmas[133X
      55.3-1 IsAdditivelyCommutative
      55.3-2 GeneratorsOfNearAdditiveMagma
      55.3-3 GeneratorsOfNearAdditiveMagmaWithZero
      55.3-4 GeneratorsOfNearAdditiveGroup
      55.3-5 AdditiveNeutralElement
      55.3-6 TrivialSubnearAdditiveMagmaWithZero
    55.4 [33X[0;0YOperations for (Near-)Additive Magmas[133X
      55.4-1 [33X[0;0YClosureNearAdditiveGroup[133X
      55.4-2 ShowAdditionTable
  56 [33X[0;0YRings[133X
    56.1 [33X[0;0YGenerating Rings[133X
      56.1-1 IsRing
      56.1-2 [33X[0;0YRing[133X
      56.1-3 [33X[0;0YDefaultRing[133X
      56.1-4 RingByGenerators
      56.1-5 DefaultRingByGenerators
      56.1-6 GeneratorsOfRing
      56.1-7 Subring
      56.1-8 [33X[0;0YClosureRing[133X
      56.1-9 Quotient
    56.2 [33X[0;0YIdeals in Rings[133X
      56.2-1 TwoSidedIdeal
      56.2-2 TwoSidedIdealNC
      56.2-3 IsTwoSidedIdeal
      56.2-4 TwoSidedIdealByGenerators
      56.2-5 LeftIdealByGenerators
      56.2-6 RightIdealByGenerators
      56.2-7 GeneratorsOfTwoSidedIdeal
      56.2-8 GeneratorsOfLeftIdeal
      56.2-9 GeneratorsOfRightIdeal
      56.2-10 LeftActingRingOfIdeal
      56.2-11 AsLeftIdeal
    56.3 [33X[0;0YRings With One[133X
      56.3-1 IsRingWithOne
      56.3-2 [33X[0;0YRingWithOne[133X
      56.3-3 RingWithOneByGenerators
      56.3-4 GeneratorsOfRingWithOne
      56.3-5 SubringWithOne
    56.4 [33X[0;0YProperties of Rings[133X
      56.4-1 IsIntegralRing
      56.4-2 IsUniqueFactorizationRing
      56.4-3 IsLDistributive
      56.4-4 IsRDistributive
      56.4-5 IsDistributive
      56.4-6 IsAnticommutative
      56.4-7 IsZeroSquaredRing
      56.4-8 IsJacobianRing
    56.5 [33X[0;0YUnits and Factorizations[133X
      56.5-1 IsUnit
      56.5-2 Units
      56.5-3 IsAssociated
      56.5-4 Associates
      56.5-5 StandardAssociate
      56.5-6 StandardAssociateUnit
      56.5-7 IsIrreducibleRingElement
      56.5-8 IsPrime
      56.5-9 Factors
      56.5-10 PadicValuation
    56.6 [33X[0;0YEuclidean Rings[133X
      56.6-1 IsEuclideanRing
      56.6-2 EuclideanDegree
      56.6-3 EuclideanQuotient
      56.6-4 EuclideanRemainder
      56.6-5 QuotientRemainder
    56.7 [33X[0;0YGcd and Lcm[133X
      56.7-1 [33X[0;0YGcd[133X
      56.7-2 GcdOp
      56.7-3 [33X[0;0YGcdRepresentation[133X
      56.7-4 GcdRepresentationOp
      56.7-5 ShowGcd
      56.7-6 [33X[0;0YLcm[133X
      56.7-7 LcmOp
      56.7-8 QuotientMod
      56.7-9 PowerMod
      56.7-10 InterpolatedPolynomial
    56.8 [33X[0;0YHomomorphisms of Rings[133X
      56.8-1 RingGeneralMappingByImages
      56.8-2 RingHomomorphismByImages
      56.8-3 RingHomomorphismByImagesNC
      56.8-4 NaturalHomomorphismByIdeal
  57 [33X[0;0YModules[133X
    57.1 [33X[0;0YGenerating modules[133X
      57.1-1 IsLeftOperatorAdditiveGroup
      57.1-2 IsLeftModule
      57.1-3 GeneratorsOfLeftOperatorAdditiveGroup
      57.1-4 GeneratorsOfLeftModule
      57.1-5 AsLeftModule
      57.1-6 IsRightOperatorAdditiveGroup
      57.1-7 IsRightModule
      57.1-8 GeneratorsOfRightOperatorAdditiveGroup
      57.1-9 GeneratorsOfRightModule
      57.1-10 LeftModuleByGenerators
      57.1-11 LeftActingDomain
    57.2 [33X[0;0YSubmodules[133X
      57.2-1 Submodule
      57.2-2 SubmoduleNC
      57.2-3 ClosureLeftModule
      57.2-4 TrivialSubmodule
    57.3 [33X[0;0YFree Modules[133X
      57.3-1 IsFreeLeftModule
      57.3-2 FreeLeftModule
      57.3-3 Dimension
      57.3-4 IsFiniteDimensional
      57.3-5 UseBasis
      57.3-6 IsRowModule
      57.3-7 IsMatrixModule
      57.3-8 IsFullRowModule
      57.3-9 FullRowModule
      57.3-10 IsFullMatrixModule
      57.3-11 FullMatrixModule
  58 [33X[0;0YFields and Division Rings[133X
    58.1 [33X[0;0YGenerating Fields[133X
      58.1-1 IsDivisionRing
      58.1-2 IsField
      58.1-3 Field
      58.1-4 DefaultField
      58.1-5 DefaultFieldByGenerators
      58.1-6 GeneratorsOfDivisionRing
      58.1-7 GeneratorsOfField
      58.1-8 DivisionRingByGenerators
      58.1-9 AsDivisionRing
    58.2 [33X[0;0YSubfields of Fields[133X
      58.2-1 Subfield
      58.2-2 FieldOverItselfByGenerators
      58.2-3 PrimitiveElement
      58.2-4 PrimeField
      58.2-5 IsPrimeField
      58.2-6 DegreeOverPrimeField
      58.2-7 DefiningPolynomial
      58.2-8 RootOfDefiningPolynomial
      58.2-9 FieldExtension
      58.2-10 Subfields
    58.3 [33X[0;0YGalois Action[133X
      58.3-1 GaloisGroup
      58.3-2 MinimalPolynomial
      58.3-3 TracePolynomial
      58.3-4 Norm
      58.3-5 [33X[0;0YTraces of field elements and matrices[133X
      58.3-6 Conjugates
      58.3-7 NormalBase
  59 [33X[0;0YFinite Fields[133X
    59.1 [33X[0;0YFinite Field Elements[133X
      59.1-1 IsFFE
      59.1-2 Z
      59.1-3 IsLexOrderedFFE
    59.2 [33X[0;0YOperations for Finite Field Elements[133X
      59.2-1 DegreeFFE
      59.2-2 LogFFE
      59.2-3 IntFFE
      59.2-4 IntFFESymm
      59.2-5 IntVecFFE
      59.2-6 AsInternalFFE
    59.3 [33X[0;0YCreating Finite Fields[133X
      59.3-1 DefaultField
      59.3-2 GaloisField
      59.3-3 PrimitiveRoot
    59.4 [33X[0;0YFrobenius Automorphisms[133X
      59.4-1 FrobeniusAutomorphism
    59.5 [33X[0;0YConway Polynomials[133X
      59.5-1 ConwayPolynomial
      59.5-2 IsCheapConwayPolynomial
      59.5-3 RandomPrimitivePolynomial
    59.6 [33X[0;0YPrinting, Viewing and Displaying Finite Field Elements[133X
      59.6-1 ViewObj
  60 [33X[0;0YAbelian Number Fields[133X
    60.1 [33X[0;0YConstruction of Abelian Number Fields[133X
      60.1-1 CyclotomicField
      60.1-2 AbelianNumberField
      60.1-3 GaussianRationals
    60.2 [33X[0;0YOperations for Abelian Number Fields[133X
      60.2-1 Factors
      60.2-2 IsNumberField
      60.2-3 IsAbelianNumberField
      60.2-4 IsCyclotomicField
      60.2-5 GaloisStabilizer
    60.3 [33X[0;0YIntegral Bases of Abelian Number Fields[133X
      60.3-1 ZumbroichBase
      60.3-2 LenstraBase
    60.4 [33X[0;0YGalois Groups of Abelian Number Fields[133X
      60.4-1 GaloisGroup
      60.4-2 ANFAutomorphism
    60.5 [33X[0;0YGaussians[133X
      60.5-1 GaussianIntegers
      60.5-2 IsGaussianIntegers
  61 [33X[0;0YVector Spaces[133X
    61.1 [33X[0;0YIsLeftVectorSpace (Filter)[133X
      61.1-1 IsLeftVectorSpace
    61.2 [33X[0;0YConstructing Vector Spaces[133X
      61.2-1 VectorSpace
      61.2-2 Subspace
      61.2-3 AsVectorSpace
      61.2-4 AsSubspace
    61.3 [33X[0;0YOperations and Attributes for Vector Spaces[133X
      61.3-1 GeneratorsOfLeftVectorSpace
      61.3-2 TrivialSubspace
    61.4 [33X[0;0YDomains of Subspaces of Vector Spaces[133X
      61.4-1 Subspaces
      61.4-2 IsSubspacesVectorSpace
    61.5 [33X[0;0YBases of Vector Spaces[133X
      61.5-1 IsBasis
      61.5-2 Basis
      61.5-3 CanonicalBasis
      61.5-4 RelativeBasis
    61.6 [33X[0;0YOperations for Vector Space Bases[133X
      61.6-1 BasisVectors
      61.6-2 UnderlyingLeftModule
      61.6-3 Coefficients
      61.6-4 LinearCombination
      61.6-5 EnumeratorByBasis
      61.6-6 IteratorByBasis
    61.7 [33X[0;0YOperations for Special Kinds of Bases[133X
      61.7-1 IsCanonicalBasis
      61.7-2 IsIntegralBasis
      61.7-3 IsNormalBasis
    61.8 [33X[0;0YMutable Bases[133X
      61.8-1 IsMutableBasis
      61.8-2 MutableBasis
      61.8-3 NrBasisVectors
      61.8-4 ImmutableBasis
      61.8-5 IsContainedInSpan
      61.8-6 CloseMutableBasis
    61.9 [33X[0;0YRow and Matrix Spaces[133X
      61.9-1 IsRowSpace
      61.9-2 IsMatrixSpace
      61.9-3 IsGaussianSpace
      61.9-4 FullRowSpace
      61.9-5 FullMatrixSpace
      61.9-6 DimensionOfVectors
      61.9-7 IsSemiEchelonized
      61.9-8 SemiEchelonBasis
      61.9-9 IsCanonicalBasisFullRowModule
      61.9-10 IsCanonicalBasisFullMatrixModule
      61.9-11 NormedRowVectors
      61.9-12 SiftedVector
    61.10 [33X[0;0YVector Space Homomorphisms[133X
      61.10-1 LeftModuleGeneralMappingByImages
      61.10-2 LeftModuleHomomorphismByImages
      61.10-3 LeftModuleHomomorphismByMatrix
      61.10-4 NaturalHomomorphismBySubspace
      61.10-5 Hom
      61.10-6 End
      61.10-7 IsFullHomModule
      61.10-8 IsPseudoCanonicalBasisFullHomModule
      61.10-9 IsLinearMappingsModule
    61.11 [33X[0;0YVector Spaces Handled By Nice Bases[133X
      61.11-1 NiceFreeLeftModule
      61.11-2 NiceVector
      61.11-3 NiceFreeLeftModuleInfo
      61.11-4 NiceBasis
      61.11-5 IsBasisByNiceBasis
      61.11-6 IsHandledByNiceBasis
    61.12 [33X[0;0YHow to Implement New Kinds of Vector Spaces[133X
      61.12-1 DeclareHandlingByNiceBasis
      61.12-2 NiceBasisFiltersInfo
      61.12-3 CheckForHandlingByNiceBasis
  62 [33X[0;0YAlgebras[133X
    62.1 [33X[0;0YInfoAlgebra (Info Class)[133X
      62.1-1 InfoAlgebra
    62.2 [33X[0;0YConstructing Algebras by Generators[133X
      62.2-1 Algebra
      62.2-2 AlgebraWithOne
    62.3 [33X[0;0YConstructing Algebras as Free Algebras[133X
      62.3-1 FreeAlgebra
      62.3-2 FreeAlgebraWithOne
      62.3-3 FreeAssociativeAlgebra
      62.3-4 FreeAssociativeAlgebraWithOne
    62.4 [33X[0;0YConstructing Algebras by Structure Constants[133X
      62.4-1 AlgebraByStructureConstants
      62.4-2 StructureConstantsTable
      62.4-3 EmptySCTable
      62.4-4 SetEntrySCTable
      62.4-5 GapInputSCTable
      62.4-6 TestJacobi
      62.4-7 IdentityFromSCTable
      62.4-8 QuotientFromSCTable
    62.5 [33X[0;0YSome Special Algebras[133X
      62.5-1 QuaternionAlgebra
      62.5-2 ComplexificationQuat
      62.5-3 OctaveAlgebra
      62.5-4 FullMatrixAlgebra
      62.5-5 NullAlgebra
    62.6 [33X[0;0YSubalgebras[133X
      62.6-1 Subalgebra
      62.6-2 SubalgebraNC
      62.6-3 SubalgebraWithOne
      62.6-4 SubalgebraWithOneNC
      62.6-5 TrivialSubalgebra
    62.7 [33X[0;0YIdeals[133X
    62.8 [33X[0;0YCategories and Properties of Algebras[133X
      62.8-1 IsFLMLOR
      62.8-2 IsFLMLORWithOne
      62.8-3 IsAlgebra
      62.8-4 IsAlgebraWithOne
      62.8-5 IsLieAlgebra
      62.8-6 IsSimpleAlgebra
      62.8-7 IsFiniteDimensional
      62.8-8 IsQuaternion
    62.9 [33X[0;0YAttributes and Operations for Algebras[133X
      62.9-1 GeneratorsOfAlgebra
      62.9-2 GeneratorsOfAlgebraWithOne
      62.9-3 ProductSpace
      62.9-4 PowerSubalgebraSeries
      62.9-5 AdjointBasis
      62.9-6 IndicesOfAdjointBasis
      62.9-7 AsAlgebra
      62.9-8 AsAlgebraWithOne
      62.9-9 AsSubalgebra
      62.9-10 AsSubalgebraWithOne
      62.9-11 MutableBasisOfClosureUnderAction
      62.9-12 MutableBasisOfNonassociativeAlgebra
      62.9-13 MutableBasisOfIdealInNonassociativeAlgebra
      62.9-14 DirectSumOfAlgebras
      62.9-15 FullMatrixAlgebraCentralizer
      62.9-16 RadicalOfAlgebra
      62.9-17 CentralIdempotentsOfAlgebra
      62.9-18 DirectSumDecomposition
      62.9-19 LeviMalcevDecomposition
      62.9-20 Grading
    62.10 [33X[0;0YHomomorphisms of Algebras[133X
      62.10-1 AlgebraGeneralMappingByImages
      62.10-2 AlgebraHomomorphismByImages
      62.10-3 AlgebraHomomorphismByImagesNC
      62.10-4 AlgebraWithOneGeneralMappingByImages
      62.10-5 AlgebraWithOneHomomorphismByImages
      62.10-6 AlgebraWithOneHomomorphismByImagesNC
      62.10-7 NaturalHomomorphismByIdeal
      62.10-8 OperationAlgebraHomomorphism
      62.10-9 NiceAlgebraMonomorphism
      62.10-10 IsomorphismFpAlgebra
      62.10-11 IsomorphismMatrixAlgebra
      62.10-12 IsomorphismSCAlgebra
      62.10-13 RepresentativeLinearOperation
    62.11 [33X[0;0YRepresentations of Algebras[133X
      62.11-1 LeftAlgebraModuleByGenerators
      62.11-2 RightAlgebraModuleByGenerators
      62.11-3 BiAlgebraModuleByGenerators
      62.11-4 LeftAlgebraModule
      62.11-5 RightAlgebraModule
      62.11-6 BiAlgebraModule
      62.11-7 GeneratorsOfAlgebraModule
      62.11-8 IsAlgebraModuleElement
      62.11-9 IsLeftAlgebraModuleElement
      62.11-10 IsRightAlgebraModuleElement
      62.11-11 LeftActingAlgebra
      62.11-12 RightActingAlgebra
      62.11-13 ActingAlgebra
      62.11-14 IsBasisOfAlgebraModuleElementSpace
      62.11-15 MatrixOfAction
      62.11-16 SubAlgebraModule
      62.11-17 LeftModuleByHomomorphismToMatAlg
      62.11-18 RightModuleByHomomorphismToMatAlg
      62.11-19 AdjointModule
      62.11-20 FaithfulModule
      62.11-21 ModuleByRestriction
      62.11-22 NaturalHomomorphismBySubAlgebraModule
      62.11-23 DirectSumOfAlgebraModules
      62.11-24 TranslatorSubalgebra
  63 [33X[0;0YFinitely Presented Algebras[133X
  64 [33X[0;0YLie Algebras[133X
    64.1 [33X[0;0YLie Objects[133X
      64.1-1 LieObject
      64.1-2 IsLieObject
      64.1-3 LieFamily
      64.1-4 UnderlyingFamily
      64.1-5 UnderlyingRingElement
    64.2 [33X[0;0YConstructing Lie algebras[133X
      64.2-1 LieAlgebraByStructureConstants
      64.2-2 RestrictedLieAlgebraByStructureConstants
      64.2-3 LieAlgebra
      64.2-4 FreeLieAlgebra
      64.2-5 FullMatrixLieAlgebra
      64.2-6 RightDerivations
      64.2-7 SimpleLieAlgebra
    64.3 [33X[0;0YDistinguished Subalgebras[133X
      64.3-1 LieCentre
      64.3-2 LieCentralizer
      64.3-3 LieNormalizer
      64.3-4 LieDerivedSubalgebra
      64.3-5 LieNilRadical
      64.3-6 LieSolvableRadical
      64.3-7 CartanSubalgebra
    64.4 [33X[0;0YSeries of Ideals[133X
      64.4-1 LieDerivedSeries
      64.4-2 LieLowerCentralSeries
      64.4-3 LieUpperCentralSeries
    64.5 [33X[0;0YProperties of a Lie Algebra[133X
      64.5-1 IsLieAbelian
      64.5-2 IsLieNilpotent
      64.5-3 IsLieSolvable
    64.6 [33X[0;0YSemisimple Lie Algebras and Root Systems[133X
      64.6-1 SemiSimpleType
      64.6-2 ChevalleyBasis
      64.6-3 IsRootSystem
      64.6-4 IsRootSystemFromLieAlgebra
      64.6-5 RootSystem
      64.6-6 UnderlyingLieAlgebra
      64.6-7 PositiveRoots
      64.6-8 NegativeRoots
      64.6-9 PositiveRootVectors
      64.6-10 NegativeRootVectors
      64.6-11 SimpleSystem
      64.6-12 CartanMatrix
      64.6-13 BilinearFormMat
      64.6-14 CanonicalGenerators
    64.7 [33X[0;0YSemisimple Lie Algebras and Weyl Groups of Root Systems[133X
      64.7-1 IsWeylGroup
      64.7-2 SparseCartanMatrix
      64.7-3 WeylGroup
      64.7-4 ApplySimpleReflection
      64.7-5 LongestWeylWordPerm
      64.7-6 ConjugateDominantWeight
      64.7-7 WeylOrbitIterator
    64.8 [33X[0;0YRestricted Lie algebras[133X
      64.8-1 IsRestrictedLieAlgebra
      64.8-2 PthPowerImages
      64.8-3 PthPowerImage
      64.8-4 JenningsLieAlgebra
      64.8-5 PCentralLieAlgebra
      64.8-6 NaturalHomomorphismOfLieAlgebraFromNilpotentGroup
    64.9 [33X[0;0YThe Adjoint Representation[133X
      64.9-1 AdjointMatrix
      64.9-2 AdjointAssociativeAlgebra
      64.9-3 KillingMatrix
      64.9-4 KappaPerp
      64.9-5 IsNilpotentElement
      64.9-6 NonNilpotentElement
      64.9-7 FindSl2
    64.10 [33X[0;0YUniversal Enveloping Algebras[133X
      64.10-1 UniversalEnvelopingAlgebra
    64.11 [33X[0;0YFinitely Presented Lie Algebras[133X
      64.11-1 FpLieAlgebraByCartanMatrix
      64.11-2 NilpotentQuotientOfFpLieAlgebra
    64.12 [33X[0;0YModules over Lie Algebras and Their Cohomology[133X
      64.12-1 IsCochain
      64.12-2 Cochain
      64.12-3 CochainSpace
      64.12-4 ValueCochain
      64.12-5 LieCoboundaryOperator
      64.12-6 Cocycles
      64.12-7 Coboundaries
    64.13 [33X[0;0YModules over Semisimple Lie Algebras[133X
      64.13-1 DominantWeights
      64.13-2 DominantCharacter
      64.13-3 DecomposeTensorProduct
      64.13-4 DimensionOfHighestWeightModule
    64.14 [33X[0;0YAdmissible Lattices in UEA[133X
      64.14-1 IsUEALatticeElement
      64.14-2 LatticeGeneratorsInUEA
      64.14-3 ObjByExtRep
      64.14-4 IsWeightRepElement
      64.14-5 HighestWeightModule
    64.15 [33X[0;0YTensor Products and Exterior and Symmetric Powers[133X
      64.15-1 TensorProductOfAlgebraModules
      64.15-2 ExteriorPowerOfAlgebraModule
      64.15-3 SymmetricPowerOfAlgebraModule
  65 [33X[0;0YMagma Rings[133X
    65.1 [33X[0;0YFree Magma Rings[133X
      65.1-1 FreeMagmaRing
      65.1-2 GroupRing
      65.1-3 IsFreeMagmaRing
      65.1-4 IsFreeMagmaRingWithOne
      65.1-5 IsGroupRing
      65.1-6 UnderlyingMagma
      65.1-7 AugmentationIdeal
    65.2 [33X[0;0YElements of Free Magma Rings[133X
      65.2-1 IsMagmaRingObjDefaultRep
      65.2-2 IsElementOfFreeMagmaRing
      65.2-3 IsElementOfFreeMagmaRingFamily
      65.2-4 CoefficientsAndMagmaElements
      65.2-5 ZeroCoefficient
      65.2-6 ElementOfMagmaRing
    65.3 [33X[0;0YNatural Embeddings related to Magma Rings[133X
    65.4 [33X[0;0YMagma Rings modulo Relations[133X
      65.4-1 IsElementOfMagmaRingModuloRelations
      65.4-2 IsElementOfMagmaRingModuloRelationsFamily
      65.4-3 NormalizedElementOfMagmaRingModuloRelations
      65.4-4 IsMagmaRingModuloRelations
    65.5 [33X[0;0YMagma Rings modulo the Span of a Zero Element[133X
      65.5-1 IsElementOfMagmaRingModuloSpanOfZeroFamily
      65.5-2 IsMagmaRingModuloSpanOfZero
      65.5-3 MagmaRingModuloSpanOfZero
    65.6 [33X[0;0YTechnical Details about the Implementation of Magma Rings[133X
  66 [33X[0;0YPolynomials and Rational Functions[133X
    66.1 [33X[0;0YIndeterminates[133X
      66.1-1 [33X[0;0YIndeterminate[133X
      66.1-2 IndeterminateNumberOfUnivariateRationalFunction
      66.1-3 IndeterminateOfUnivariateRationalFunction
      66.1-4 IndeterminateName
      66.1-5 CIUnivPols
    66.2 [33X[0;0YOperations for Rational Functions[133X
    66.3 [33X[0;0YComparison of Rational Functions[133X
    66.4 [33X[0;0YProperties and Attributes of Rational Functions[133X
      66.4-1 IsPolynomialFunction
      66.4-2 NumeratorOfRationalFunction
      66.4-3 DenominatorOfRationalFunction
      66.4-4 IsPolynomial
      66.4-5 AsPolynomial
      66.4-6 IsUnivariateRationalFunction
      66.4-7 CoefficientsOfUnivariateRationalFunction
      66.4-8 IsUnivariatePolynomial
      66.4-9 CoefficientsOfUnivariatePolynomial
      66.4-10 IsLaurentPolynomial
      66.4-11 IsConstantRationalFunction
      66.4-12 IsPrimitivePolynomial
      66.4-13 SplittingField
    66.5 [33X[0;0YUnivariate Polynomials[133X
      66.5-1 UnivariatePolynomial
      66.5-2 UnivariatePolynomialByCoefficients
      66.5-3 DegreeOfLaurentPolynomial
      66.5-4 RootsOfPolynomial
      66.5-5 RootsOfUPol
      66.5-6 QuotRemLaurpols
      66.5-7 UnivariatenessTestRationalFunction
      66.5-8 InfoPoly
    66.6 [33X[0;0YPolynomials as Univariate Polynomials in one Indeterminate[133X
      66.6-1 DegreeIndeterminate
      66.6-2 PolynomialCoefficientsOfPolynomial
      66.6-3 LeadingCoefficient
      66.6-4 LeadingMonomial
      66.6-5 Derivative
      66.6-6 Discriminant
      66.6-7 Resultant
    66.7 [33X[0;0YMultivariate Polynomials[133X
      66.7-1 [33X[0;0YValue[133X
    66.8 [33X[0;0YMinimal Polynomials[133X
      66.8-1 MinimalPolynomial
    66.9 [33X[0;0YCyclotomic Polynomials[133X
      66.9-1 CyclotomicPolynomial
    66.10 [33X[0;0YPolynomial Factorization[133X
      66.10-1 Factors
      66.10-2 FactorsSquarefree
    66.11 [33X[0;0YPolynomials over the Rationals[133X
      66.11-1 PrimitivePolynomial
      66.11-2 PolynomialModP
      66.11-3 GaloisType
      66.11-4 ProbabilityShapes
    66.12 [33X[0;0YFactorization of Polynomials over the Rationals[133X
      66.12-1 BombieriNorm
      66.12-2 MinimizedBombieriNorm
      66.12-3 HenselBound
      66.12-4 OneFactorBound
    66.13 [33X[0;0YLaurent Polynomials[133X
      66.13-1 LaurentPolynomialByCoefficients
      66.13-2 CoefficientsOfLaurentPolynomial
      66.13-3 IndeterminateNumberOfLaurentPolynomial
    66.14 [33X[0;0YUnivariate Rational Functions[133X
      66.14-1 UnivariateRationalFunctionByCoefficients
    66.15 [33X[0;0YPolynomial Rings and Function Fields[133X
      66.15-1 [33X[0;0YPolynomialRing[133X
      66.15-2 IndeterminatesOfPolynomialRing
      66.15-3 CoefficientsRing
      66.15-4 IsPolynomialRing
      66.15-5 IsFiniteFieldPolynomialRing
      66.15-6 IsAbelianNumberFieldPolynomialRing
      66.15-7 IsRationalsPolynomialRing
      66.15-8 [33X[0;0YFunctionField[133X
      66.15-9 IsFunctionField
    66.16 [33X[0;0YUnivariate Polynomial Rings[133X
      66.16-1 [33X[0;0YUnivariatePolynomialRing[133X
      66.16-2 IsUnivariatePolynomialRing
    66.17 [33X[0;0YMonomial Orderings[133X
      66.17-1 IsMonomialOrdering
      66.17-2 LeadingMonomialOfPolynomial
      66.17-3 LeadingTermOfPolynomial
      66.17-4 LeadingCoefficientOfPolynomial
      66.17-5 MonomialComparisonFunction
      66.17-6 MonomialExtrepComparisonFun
      66.17-7 MonomialLexOrdering
      66.17-8 MonomialGrlexOrdering
      66.17-9 MonomialGrevlexOrdering
      66.17-10 EliminationOrdering
      66.17-11 PolynomialReduction
      66.17-12 PolynomialReducedRemainder
      66.17-13 PolynomialDivisionAlgorithm
      66.17-14 MonomialExtGrlexLess
    66.18 [33X[0;0YGroebner Bases[133X
      66.18-1 [33X[0;0YGroebnerBasis[133X
      66.18-2 [33X[0;0YReducedGroebnerBasis[133X
      66.18-3 StoredGroebnerBasis
      66.18-4 InfoGroebner
    66.19 [33X[0;0YRational Function Families[133X
      66.19-1 RationalFunctionsFamily
      66.19-2 IsPolynomialFunctionsFamily
      66.19-3 CoefficientsFamily
    66.20 [33X[0;0YThe Representations of Rational Functions[133X
    66.21 [33X[0;0YThe Defining Attributes of Rational Functions[133X
      66.21-1 IsRationalFunctionDefaultRep
      66.21-2 ExtRepNumeratorRatFun
      66.21-3 ExtRepDenominatorRatFun
      66.21-4 ZeroCoefficientRatFun
      66.21-5 IsPolynomialDefaultRep
      66.21-6 ExtRepPolynomialRatFun
      66.21-7 IsLaurentPolynomialDefaultRep
    66.22 [33X[0;0YCreation of Rational Functions[133X
      66.22-1 RationalFunctionByExtRep
      66.22-2 PolynomialByExtRep
      66.22-3 LaurentPolynomialByExtRep
    66.23 [33X[0;0YArithmetic for External Representations of Polynomials[133X
      66.23-1 ZippedSum
      66.23-2 ZippedProduct
      66.23-3 QuotientPolynomialsExtRep
    66.24 [33X[0;0YCancellation Tests for Rational Functions[133X
      66.24-1 RationalFunctionByExtRepWithCancellation
      66.24-2 TryGcdCancelExtRepPolynomials
      66.24-3 HeuristicCancelPolynomials
  67 [33X[0;0YAlgebraic extensions of fields[133X
    67.1 [33X[0;0YCreation of Algebraic Extensions[133X
      67.1-1 AlgebraicExtension
      67.1-2 IsAlgebraicExtension
    67.2 [33X[0;0YElements in Algebraic Extensions[133X
      67.2-1 IsAlgebraicElement
  68 [33X[0;0Yp-adic Numbers (preliminary)[133X
    68.1 [33X[0;0YPure p-adic Numbers[133X
      68.1-1 PurePadicNumberFamily
      68.1-2 PadicNumber
      68.1-3 Valuation
      68.1-4 ShiftedPadicNumber
      68.1-5 IsPurePadicNumber
      68.1-6 IsPurePadicNumberFamily
    68.2 [33X[0;0YExtensions of the p-adic Numbers[133X
      68.2-1 PadicExtensionNumberFamily
      68.2-2 PadicNumber
      68.2-3 IsPadicExtensionNumber
      68.2-4 IsPadicExtensionNumberFamily
  69 [33X[0;0YThe MeatAxe[133X
    69.1 [33X[0;0YMeatAxe Modules[133X
      69.1-1 [33X[0;0YGModuleByMats[133X
    69.2 [33X[0;0YModule Constructions[133X
      69.2-1 PermutationGModule
      69.2-2 TensorProductGModule
      69.2-3 WedgeGModule
    69.3 [33X[0;0YSelecting a Different MeatAxe[133X
      69.3-1 MTX
    69.4 [33X[0;0YAccessing a Module[133X
      69.4-1 MTX.Generators
      69.4-2 MTX.Dimension
      69.4-3 MTX.Field
    69.5 [33X[0;0YIrreducibility Tests[133X
      69.5-1 MTX.IsIrreducible
      69.5-2 MTX.IsAbsolutelyIrreducible
      69.5-3 MTX.DegreeSplittingField
    69.6 [33X[0;0YDecomposition of modules[133X
      69.6-1 MTX.IsIndecomposable
      69.6-2 MTX.Indecomposition
      69.6-3 MTX.HomogeneousComponents
    69.7 [33X[0;0YFinding Submodules[133X
      69.7-1 MTX.SubmoduleGModule
      69.7-2 MTX.ProperSubmoduleBasis
      69.7-3 MTX.BasesSubmodules
      69.7-4 MTX.BasesMinimalSubmodules
      69.7-5 MTX.BasesMaximalSubmodules
      69.7-6 MTX.BasisRadical
      69.7-7 MTX.BasisSocle
      69.7-8 MTX.BasesMinimalSupermodules
      69.7-9 MTX.BasesCompositionSeries
      69.7-10 MTX.CompositionFactors
      69.7-11 MTX.CollectedFactors
    69.8 [33X[0;0YInduced Actions[133X
      69.8-1 MTX.NormedBasisAndBaseChange
      69.8-2 MTX.InducedActionSubmodule
      69.8-3 MTX.InducedActionFactorModule
      69.8-4 MTX.InducedActionMatrix
      69.8-5 MTX.InducedAction
    69.9 [33X[0;0YModule Homomorphisms[133X
      69.9-1 MTX.BasisModuleHomomorphisms
      69.9-2 MTX.BasisModuleEndomorphisms
      69.9-3 MTX.IsomorphismModules
      69.9-4 MTX.ModuleAutomorphisms
    69.10 [33X[0;0YModule Homomorphisms for irreducible modules[133X
      69.10-1 MTX.IsEquivalent
      69.10-2 MTX.IsomorphismIrred
      69.10-3 MTX.Homomorphism
      69.10-4 MTX.Homomorphisms
      69.10-5 MTX.Distinguish
    69.11 [33X[0;0YMeatAxe Functionality for Invariant Forms[133X
      69.11-1 MTX.InvariantBilinearForm
      69.11-2 MTX.InvariantSesquilinearForm
      69.11-3 MTX.InvariantQuadraticForm
      69.11-4 MTX.BasisInOrbit
      69.11-5 MTX.OrthogonalSign
    69.12 [33X[0;0YThe Smash MeatAxe[133X
      69.12-1 SMTX.RandomIrreducibleSubGModule
      69.12-2 SMTX.GoodElementGModule
      69.12-3 SMTX.SortHomGModule
      69.12-4 SMTX.MinimalSubGModules
      69.12-5 SMTX.Setter
      69.12-6 SMTX.Getter
      69.12-7 SMTX.IrreducibilityTest
      69.12-8 SMTX.AbsoluteIrreducibilityTest
      69.12-9 SMTX.MinimalSubGModule
      69.12-10 SMTX.MatrixSum
      69.12-11 SMTX.CompleteBasis
    69.13 [33X[0;0YSmash MeatAxe Flags[133X
      69.13-1 SMTX.Subbasis
      69.13-2 SMTX.AlgEl
      69.13-3 SMTX.AlgElMat
      69.13-4 SMTX.AlgElCharPol
      69.13-5 SMTX.AlgElCharPolFac
      69.13-6 SMTX.AlgElNullspaceVec
      69.13-7 SMTX.AlgElNullspaceDimension
      69.13-8 SMTX.CentMat
      69.13-9 SMTX.CentMatMinPoly
  70 [33X[0;0YTables of Marks[133X
    70.1 [33X[0;0YMore about Tables of Marks[133X
    70.2 [33X[0;0YTable of Marks Objects in GAP[133X
    70.3 [33X[0;0YConstructing Tables of Marks[133X
      70.3-1 TableOfMarks
      70.3-2 TableOfMarksByLattice
      70.3-3 LatticeSubgroupsByTom
    70.4 [33X[0;0YPrinting Tables of Marks[133X
      70.4-1 ViewObj
      70.4-2 PrintObj
      70.4-3 Display
    70.5 [33X[0;0YSorting Tables of Marks[133X
      70.5-1 SortedTom
      70.5-2 PermutationTom
    70.6 [33X[0;0YTechnical Details about Tables of Marks[133X
      70.6-1 InfoTom
      70.6-2 IsTableOfMarks
      70.6-3 TableOfMarksFamily
      70.6-4 TableOfMarksComponents
      70.6-5 ConvertToTableOfMarks
    70.7 [33X[0;0YAttributes of Tables of Marks[133X
      70.7-1 MarksTom
      70.7-2 NrSubsTom
      70.7-3 LengthsTom
      70.7-4 ClassTypesTom
      70.7-5 ClassNamesTom
      70.7-6 FusionsTom
      70.7-7 UnderlyingGroup
      70.7-8 IdempotentsTom
      70.7-9 Identifier
      70.7-10 MatTom
      70.7-11 MoebiusTom
      70.7-12 WeightsTom
    70.8 [33X[0;0YProperties of Tables of Marks[133X
      70.8-1 IsAbelianTom
    70.9 [33X[0;0YOther Operations for Tables of Marks[133X
      70.9-1 IsInternallyConsistent
      70.9-2 DerivedSubgroupTom
      70.9-3 DerivedSubgroupsTomPossible
      70.9-4 NormalizerTom
      70.9-5 ContainedTom
      70.9-6 ContainingTom
      70.9-7 CyclicExtensionsTom
      70.9-8 DecomposedFixedPointVector
      70.9-9 EulerianFunctionByTom
      70.9-10 IntersectionsTom
      70.9-11 FactorGroupTom
      70.9-12 MaximalSubgroupsTom
      70.9-13 MinimalSupergroupsTom
    70.10 [33X[0;0YAccessing Subgroups via Tables of Marks[133X
      70.10-1 GeneratorsSubgroupsTom
      70.10-2 StraightLineProgramsTom
      70.10-3 IsTableOfMarksWithGens
      70.10-4 RepresentativeTom
    70.11 [33X[0;0YThe Interface between Tables of Marks and Character Tables[133X
      70.11-1 FusionCharTableTom
      70.11-2 PermCharsTom
    70.12 [33X[0;0YGeneric Construction of Tables of Marks[133X
      70.12-1 TableOfMarksCyclic
      70.12-2 TableOfMarksDihedral
      70.12-3 TableOfMarksFrobenius
    70.13 [33X[0;0YThe Library of Tables of Marks[133X
  71 [33X[0;0YCharacter Tables[133X
    71.1 [33X[0;0YSome Remarks about Character Theory in [5XGAP[105X[133X
    71.2 [33X[0;0YHistory of Character Theory Stuff in GAP[133X
    71.3 [33X[0;0YCreating Character Tables[133X
      71.3-1 [33X[0;0YCharacterTable[133X
      71.3-2 [33X[0;0YBrauerTable[133X
      71.3-3 CharacterTableRegular
      71.3-4 SupportedCharacterTableInfo
      71.3-5 ConvertToCharacterTable
    71.4 [33X[0;0YCharacter Table Categories[133X
      71.4-1 IsNearlyCharacterTable
      71.4-2 InfoCharacterTable
      71.4-3 NearlyCharacterTablesFamily
    71.5 [33X[0;0YConventions for Character Tables[133X
    71.6 [33X[0;0YThe Interface between Character Tables and Groups[133X
      71.6-1 UnderlyingGroup
      71.6-2 ConjugacyClasses
      71.6-3 IdentificationOfConjugacyClasses
      71.6-4 CharacterTableWithStoredGroup
      71.6-5 CompatibleConjugacyClasses
    71.7 [33X[0;0YOperators for Character Tables[133X
    71.8 [33X[0;0YAttributes and Properties for Groups and Character Tables[133X
      71.8-1 [33X[0;0YCharacterDegrees[133X
      71.8-2 [33X[0;0YIrr[133X
      71.8-3 [33X[0;0YLinearCharacters[133X
      71.8-4 [33X[0;0YOrdinaryCharacterTable[133X
      71.8-5 [33X[0;0YGroup Operations Applicable to Character Tables[133X
    71.9 [33X[0;0YAttributes and Properties only for Character Tables[133X
      71.9-1 OrdersClassRepresentatives
      71.9-2 SizesCentralizers
      71.9-3 SizesConjugacyClasses
      71.9-4 AutomorphismsOfTable
      71.9-5 [33X[0;0YUnderlyingCharacteristic[133X
      71.9-6 [33X[0;0YClass Names and Character Names[133X
      71.9-7 [33X[0;0YClass Parameters and Character Parameters[133X
      71.9-8 Identifier
      71.9-9 InfoText
      71.9-10 InverseClasses
      71.9-11 RealClasses
      71.9-12 ClassOrbit
      71.9-13 ClassRoots
    71.10 [33X[0;0YNormal Subgroups Represented by Lists of Class Positions[133X
      71.10-1 ClassPositionsOfNormalSubgroups
      71.10-2 ClassPositionsOfAgemo
      71.10-3 ClassPositionsOfCentre
      71.10-4 ClassPositionsOfDirectProductDecompositions
      71.10-5 ClassPositionsOfDerivedSubgroup
      71.10-6 ClassPositionsOfElementaryAbelianSeries
      71.10-7 ClassPositionsOfFittingSubgroup
      71.10-8 ClassPositionsOfLowerCentralSeries
      71.10-9 ClassPositionsOfUpperCentralSeries
      71.10-10 ClassPositionsOfSupersolvableResiduum
      71.10-11 ClassPositionsOfPCore
      71.10-12 ClassPositionsOfNormalClosure
    71.11 [33X[0;0YOperations Concerning Blocks[133X
      71.11-1 PrimeBlocks
      71.11-2 SameBlock
      71.11-3 BlocksInfo
      71.11-4 DecompositionMatrix
      71.11-5 LaTeXStringDecompositionMatrix
    71.12 [33X[0;0YOther Operations for Character Tables[133X
      71.12-1 Index
      71.12-2 IsInternallyConsistent
      71.12-3 IsPSolvableCharacterTable
      71.12-4 IsClassFusionOfNormalSubgroup
      71.12-5 Indicator
      71.12-6 NrPolyhedralSubgroups
      71.12-7 ClassMultiplicationCoefficient
      71.12-8 ClassStructureCharTable
      71.12-9 MatClassMultCoeffsCharTable
    71.13 [33X[0;0YPrinting Character Tables[133X
      71.13-1 ViewObj
      71.13-2 PrintObj
      71.13-3 Display
      71.13-4 DisplayOptions
      71.13-5 PrintCharacterTable
    71.14 [33X[0;0YComputing the Irreducible Characters of a Group[133X
      71.14-1 IrrDixonSchneider
      71.14-2 IrrConlon
      71.14-3 IrrBaumClausen
      71.14-4 IrreducibleRepresentations
      71.14-5 IrreducibleRepresentationsDixon
    71.15 [33X[0;0YRepresentations Given by Modules[133X
      71.15-1 IrreducibleModules
      71.15-2 AbsolutelyIrreducibleModules
      71.15-3 RegularModule
    71.16 [33X[0;0YThe Dixon-Schneider Algorithm[133X
    71.17 [33X[0;0YAdvanced Methods for Dixon-Schneider Calculations[133X
      71.17-1 DixonRecord
      71.17-2 DixonInit
      71.17-3 DixontinI
      71.17-4 DixonSplit
      71.17-5 BestSplittingMatrix
      71.17-6 DxIncludeIrreducibles
      71.17-7 SplitCharacters
      71.17-8 IsDxLargeGroup
    71.18 [33X[0;0YComponents of a Dixon Record[133X
    71.19 [33X[0;0YAn Example of Advanced Dixon-Schneider Calculations[133X
    71.20 [33X[0;0YConstructing Character Tables from Others[133X
      71.20-1 CharacterTableDirectProduct
      71.20-2 FactorsOfDirectProduct
      71.20-3 CharacterTableFactorGroup
      71.20-4 CharacterTableIsoclinic
      71.20-5 CharacterTableWreathSymmetric
    71.21 [33X[0;0YSorted Character Tables[133X
      71.21-1 CharacterTableWithSortedCharacters
      71.21-2 SortedCharacters
      71.21-3 CharacterTableWithSortedClasses
      71.21-4 SortedCharacterTable
      71.21-5 ClassPermutation
    71.22 [33X[0;0YAutomorphisms and Equivalence of Character Tables[133X
      71.22-1 MatrixAutomorphisms
      71.22-2 TableAutomorphisms
      71.22-3 TransformingPermutations
      71.22-4 TransformingPermutationsCharacterTables
      71.22-5 FamiliesOfRows
    71.23 [33X[0;0YStoring Normal Subgroup Information[133X
      71.23-1 NormalSubgroupClassesInfo
      71.23-2 ClassPositionsOfNormalSubgroup
      71.23-3 NormalSubgroupClasses
      71.23-4 FactorGroupNormalSubgroupClasses
  72 [33X[0;0YClass Functions[133X
    72.1 [33X[0;0YWhy Class Functions?[133X
      72.1-1 IsClassFunction
    72.2 [33X[0;0YBasic Operations for Class Functions[133X
      72.2-1 UnderlyingCharacterTable
      72.2-2 ValuesOfClassFunction
    72.3 [33X[0;0YComparison of Class Functions[133X
    72.4 [33X[0;0YArithmetic Operations for Class Functions[133X
      72.4-1 Characteristic
      72.4-2 ComplexConjugate
      72.4-3 Order
    72.5 [33X[0;0YPrinting Class Functions[133X
      72.5-1 ViewObj
      72.5-2 PrintObj
      72.5-3 Display
    72.6 [33X[0;0YCreating Class Functions from Values Lists[133X
      72.6-1 ClassFunction
      72.6-2 VirtualCharacter
      72.6-3 Character
      72.6-4 ClassFunctionSameType
    72.7 [33X[0;0YCreating Class Functions using Groups[133X
      72.7-1 [33X[0;0YTrivialCharacter[133X
      72.7-2 NaturalCharacter
      72.7-3 [33X[0;0YPermutationCharacter[133X
    72.8 [33X[0;0YOperations for Class Functions[133X
      72.8-1 IsCharacter
      72.8-2 IsVirtualCharacter
      72.8-3 IsIrreducibleCharacter
      72.8-4 DegreeOfCharacter
      72.8-5 ScalarProduct
      72.8-6 MatScalarProducts
      72.8-7 Norm
      72.8-8 ConstituentsOfCharacter
      72.8-9 KernelOfCharacter
      72.8-10 ClassPositionsOfKernel
      72.8-11 CentreOfCharacter
      72.8-12 ClassPositionsOfCentre
      72.8-13 InertiaSubgroup
      72.8-14 CycleStructureClass
      72.8-15 IsTransitive
      72.8-16 Transitivity
      72.8-17 CentralCharacter
      72.8-18 DeterminantOfCharacter
      72.8-19 EigenvaluesChar
      72.8-20 Tensored
    72.9 [33X[0;0YRestricted and Induced Class Functions[133X
      72.9-1 RestrictedClassFunction
      72.9-2 RestrictedClassFunctions
      72.9-3 [33X[0;0YInducedClassFunction[133X
      72.9-4 InducedClassFunctions
      72.9-5 InducedClassFunctionsByFusionMap
      72.9-6 InducedCyclic
    72.10 [33X[0;0YReducing Virtual Characters[133X
      72.10-1 ReducedClassFunctions
      72.10-2 ReducedCharacters
      72.10-3 IrreducibleDifferences
      72.10-4 LLL
      72.10-5 Extract
      72.10-6 OrthogonalEmbeddingsSpecialDimension
      72.10-7 Decreased
      72.10-8 DnLattice
      72.10-9 DnLatticeIterative
    72.11 [33X[0;0YSymmetrizations of Class Functions[133X
      72.11-1 Symmetrizations
      72.11-2 SymmetricParts
      72.11-3 AntiSymmetricParts
      72.11-4 OrthogonalComponents
      72.11-5 SymplecticComponents
    72.12 [33X[0;0YMolien Series[133X
      72.12-1 MolienSeries
      72.12-2 MolienSeriesInfo
      72.12-3 ValueMolienSeries
      72.12-4 MolienSeriesWithGivenDenominator
    72.13 [33X[0;0YPossible Permutation Characters[133X
      72.13-1 PermCharInfo
      72.13-2 PermCharInfoRelative
    72.14 [33X[0;0YComputing Possible Permutation Characters[133X
      72.14-1 PermChars
      72.14-2 [33X[0;0YTestPerm1, ..., TestPerm5[133X
      72.14-3 PermBounds
      72.14-4 PermComb
      72.14-5 Inequalities
    72.15 [33X[0;0YOperations for Brauer Characters[133X
      72.15-1 FrobeniusCharacterValue
      72.15-2 BrauerCharacterValue
      72.15-3 SizeOfFieldOfDefinition
      72.15-4 RealizableBrauerCharacters
    72.16 [33X[0;0YDomains Generated by Class Functions[133X
  73 [33X[0;0YMaps Concerning Character Tables[133X
    73.1 [33X[0;0YPower Maps[133X
      73.1-1 PowerMap
      73.1-2 PossiblePowerMaps
      73.1-3 ElementOrdersPowerMap
      73.1-4 PowerMapByComposition
    73.2 [33X[0;0YOrbits on Sets of Possible Power Maps[133X
      73.2-1 OrbitPowerMaps
      73.2-2 RepresentativesPowerMaps
    73.3 [33X[0;0YClass Fusions between Character Tables[133X
      73.3-1 [33X[0;0YFusionConjugacyClasses[133X
      73.3-2 ComputedClassFusions
      73.3-3 GetFusionMap
      73.3-4 StoreFusion
      73.3-5 NamesOfFusionSources
      73.3-6 PossibleClassFusions
      73.3-7 ConsiderStructureConstants
    73.4 [33X[0;0YOrbits on Sets of Possible Class Fusions[133X
      73.4-1 OrbitFusions
      73.4-2 RepresentativesFusions
    73.5 [33X[0;0YParametrized Maps[133X
      73.5-1 CompositionMaps
      73.5-2 InverseMap
      73.5-3 ProjectionMap
      73.5-4 Indirected
      73.5-5 Parametrized
      73.5-6 ContainedMaps
      73.5-7 UpdateMap
      73.5-8 MeetMaps
      73.5-9 CommutativeDiagram
      73.5-10 CheckFixedPoints
      73.5-11 TransferDiagram
      73.5-12 TestConsistencyMaps
      73.5-13 Indeterminateness
      73.5-14 PrintAmbiguity
      73.5-15 ContainedSpecialVectors
      73.5-16 CollapsedMat
      73.5-17 ContainedDecomposables
    73.6 [33X[0;0YSubroutines for the Construction of Power Maps[133X
      73.6-1 InitPowerMap
      73.6-2 Congruences
      73.6-3 ConsiderKernels
      73.6-4 ConsiderSmallerPowerMaps
      73.6-5 MinusCharacter
      73.6-6 PowerMapsAllowedBySymmetrizations
    73.7 [33X[0;0YSubroutines for the Construction of Class Fusions[133X
      73.7-1 InitFusion
      73.7-2 CheckPermChar
      73.7-3 ConsiderTableAutomorphisms
      73.7-4 FusionsAllowedByRestrictions
  74 [33X[0;0YUnknowns[133X
    74.1 [33X[0;0YMore about Unknowns[133X
      74.1-1 Unknown
      74.1-2 LargestUnknown
      74.1-3 IsUnknown
      74.1-4 [33X[0;0YComparison of Unknowns[133X
      74.1-5 [33X[0;0YArithmetical Operations for Unknowns[133X
  75 [33X[0;0YMonomiality Questions[133X
    75.1 [33X[0;0YInfoMonomial (Info Class)[133X
      75.1-1 InfoMonomial
    75.2 [33X[0;0YCharacter Degrees and Derived Length[133X
      75.2-1 Alpha
      75.2-2 Delta
      75.2-3 [33X[0;0YIsBergerCondition[133X
    75.3 [33X[0;0YPrimitivity of Characters[133X
      75.3-1 TestHomogeneous
      75.3-2 IsPrimitiveCharacter
      75.3-3 TestQuasiPrimitive
      75.3-4 TestInducedFromNormalSubgroup
    75.4 [33X[0;0YTesting Monomiality[133X
      75.4-1 [33X[0;0YTestMonomial[133X
      75.4-2 TestMonomialUseLattice
      75.4-3 IsMonomialNumber
      75.4-4 [33X[0;0YTestMonomialQuick[133X
      75.4-5 [33X[0;0YTestSubnormallyMonomial[133X
      75.4-6 [33X[0;0YTestRelativelySM[133X
    75.5 [33X[0;0YMinimal Nonmonomial Groups[133X
      75.5-1 IsMinimalNonmonomial
      75.5-2 MinimalNonmonomialGroup
  76 [33X[0;0YGAP Packages[133X
    76.1 [33X[0;0YInstalling a GAP Package[133X
    76.2 [33X[0;0YLoading a GAP Package[133X
      76.2-1 LoadPackage
      76.2-2 SetPackagePath
      76.2-3 ExtendRootDirectories
      76.2-4 DisplayPackageLoadingLog
    76.3 [33X[0;0YFunctions for GAP Packages[133X
      76.3-1 ReadPackage
      76.3-2 TestPackageAvailability
      76.3-3 InstalledPackageVersion
      76.3-4 DirectoriesPackageLibrary
      76.3-5 DirectoriesPackagePrograms
      76.3-6 CompareVersionNumbers
      76.3-7 IsPackageMarkedForLoading
      76.3-8 DeclareAutoreadableVariables
      76.3-9 [33X[0;0YKernel modules in [5XGAP[105X packages[133X
      76.3-10 LoadDynamicModule
      76.3-11 [33X[0;0YThe PackageInfo.g File[133X
      76.3-12 ValidatePackageInfo
      76.3-13 ShowPackageVariables
      76.3-14 BibEntry
  77 [33X[0;0YReplaced and Removed Command Names[133X
    77.1 [33X[0;0YGroup Actions – Name Changes[133X
    77.2 [33X[0;0YPackage Interface – Obsolete Functions and Name Changes[133X
    77.3 [33X[0;0YNormal Forms of Integer Matrices – Name Changes[133X
    77.4 [33X[0;0YMiscellaneous Name Changes or Removed Names[133X
    77.5 [33X[0;0YThe former .gaprc file[133X
  78 [33X[0;0YMethod Selection[133X
    78.1 [33X[0;0YOperations and Methods[133X
    78.2 [33X[0;0YMethod Installation[133X
      78.2-1 InstallMethod
      78.2-2 InstallOtherMethod
    78.3 [33X[0;0YApplicable Methods and Method Selection[133X
    78.4 [33X[0;0YPartial Methods[133X
      78.4-1 TryNextMethod
    78.5 [33X[0;0YRedispatching[133X
      78.5-1 RedispatchOnCondition
    78.6 [33X[0;0YImmediate Methods[133X
      78.6-1 InstallImmediateMethod
    78.7 [33X[0;0YLogical Implications[133X
      78.7-1 InstallTrueMethod
    78.8 [33X[0;0YOperations and Mathematical Terms[133X
  79 [33X[0;0YCreating New Objects[133X
    79.1 [33X[0;0YCreating Categories[133X
      79.1-1 NewCategory
      79.1-2 CategoryFamily
    79.2 [33X[0;0YCreating Representations[133X
      79.2-1 NewRepresentation
    79.3 [33X[0;0YCreating Attributes and Properties[133X
      79.3-1 NewAttribute
      79.3-2 NewProperty
    79.4 [33X[0;0YCreating Other Filters[133X
      79.4-1 NewFilter
      79.4-2 SetFilterObj
      79.4-3 ResetFilterObj
    79.5 [33X[0;0YCreating Operations[133X
      79.5-1 NewOperation
    79.6 [33X[0;0YCreating Families[133X
      79.6-1 NewFamily
    79.7 [33X[0;0YCreating Types[133X
      79.7-1 NewType
    79.8 [33X[0;0YCreating Objects[133X
      79.8-1 Objectify
      79.8-2 ObjectifyWithAttributes
    79.9 [33X[0;0YComponent Objects[133X
      79.9-1 NamesOfComponents
    79.10 [33X[0;0YPositional Objects[133X
    79.11 [33X[0;0YImplementing New List Objects[133X
    79.12 [33X[0;0YExample – Constructing Enumerators[133X
    79.13 [33X[0;0YExample – Constructing Iterators[133X
    79.14 [33X[0;0YArithmetic Issues in the Implementation of New Kinds of Lists[133X
    79.15 [33X[0;0YExternal Representation[133X
      79.15-1 ExtRepOfObj
    79.16 [33X[0;0YMutability and Copying[133X
    79.17 [33X[0;0YGlobal Variables in the Library[133X
      79.17-1 DeclareCategory
      79.17-2 DeclareRepresentation
      79.17-3 DeclareAttribute
      79.17-4 DeclareProperty
      79.17-5 DeclareFilter
      79.17-6 DeclareOperation
      79.17-7 DeclareGlobalFunction
      79.17-8 DeclareGlobalVariable
      79.17-9 InstallValue
      79.17-10 DeclareSynonym
      79.17-11 FlushCaches
    79.18 [33X[0;0YDeclaration and Implementation Part[133X
  80 [33X[0;0YExamples of Extending the System[133X
    80.1 [33X[0;0YAddition of a Method[133X
    80.2 [33X[0;0YExtending the Range of Definition of an Existing Operation[133X
    80.3 [33X[0;0YEnforcing Property Tests[133X
    80.4 [33X[0;0YAdding a new Operation[133X
    80.5 [33X[0;0YAdding a new Attribute[133X
    80.6 [33X[0;0YAdding a new Representation[133X
    80.7 [33X[0;0YComponents versus Attributes[133X
    80.8 [33X[0;0YAdding new Concepts[133X
      80.8-1 [33X[0;0YExample: M-groups[133X
      80.8-2 [33X[0;0YExample: Groups with a word length[133X
      80.8-3 [33X[0;0YExample: Groups with a decomposition as semidirect product[133X
    80.9 [33X[0;0YCreating Own Arithmetic Objects[133X
      80.9-1 ArithmeticElementCreator
      80.9-2 [33X[0;0YExample: ArithmeticElementCreator[133X
  81 [33X[0;0YAn Example – Residue Class Rings[133X
    81.1 [33X[0;0YA First Attempt to Implement Elements of Residue Class Rings[133X
    81.2 [33X[0;0YWhy Proceed in a Different Way?[133X
    81.3 [33X[0;0YA Second Attempt to Implement Elements of Residue Class Rings[133X
    81.4 [33X[0;0YCompatibility of Residue Class Rings with Prime Fields[133X
    81.5 [33X[0;0YFurther Improvements in Implementing Residue Class Rings[133X
  82 [33X[0;0YAn Example – Designing Arithmetic Operations[133X
    82.1 [33X[0;0YNew Arithmetic Operations vs. New Objects[133X
    82.2 [33X[0;0YDesigning new Multiplicative Objects[133X
  83 [33X[0;0YLibrary Files[133X
    83.1 [33X[0;0YFile Types[133X
    83.2 [33X[0;0YFinding Implementations in the Library[133X
    83.3 [33X[0;0YUndocumented Variables[133X
  84 [33X[0;0YInterface to the GAP Help System[133X
    84.1 [33X[0;0YInstalling and Removing a Help Book[133X
      84.1-1 HELP_ADD_BOOK
      84.1-2 HELP_REMOVE_BOOK
    84.2 [33X[0;0YThe manual.six File[133X
    84.3 [33X[0;0YThe Help Book Handler[133X
    84.4 [33X[0;0YIntroducing new Viewer for the Online Help[133X
      84.4-1 HELP_VIEWER_INFO
  85 [33X[0;0YFunction-Operation-Attribute Triples[133X
    85.1 [33X[0;0YKey Dependent Operations[133X
      85.1-1 KeyDependentOperation
    85.2 [33X[0;0YIn Parent Attributes[133X
      85.2-1 InParentFOA
    85.3 [33X[0;0YOperation Functions[133X
      85.3-1 OrbitsishOperation
      85.3-2 OrbitishFO
      85.3-3 [33X[0;0YExample: Orbit and OrbitOp[133X
  86 [33X[0;0YWeak Pointers[133X
    86.1 [33X[0;0YWeak Pointer Objects[133X
      86.1-1 WeakPointerObj
    86.2 [33X[0;0YLow Level Access Functions for Weak Pointer Objects[133X
      86.2-1 SetElmWPObj
    86.3 [33X[0;0YAccessing Weak Pointer Objects as Lists[133X
    86.4 [33X[0;0YCopying Weak Pointer Objects[133X
    86.5 [33X[0;0YThe GASMAN Interface for Weak Pointer Objects[133X
  87 [33X[0;0YMore about Stabilizer Chains[133X
    87.1 [33X[0;0YGeneralized Conjugation Technique[133X
    87.2 [33X[0;0YThe General Backtrack Algorithm with Ordered Partitions[133X
      87.2-1 [33X[0;0YInternal representation of ordered partitions[133X
      87.2-2 [33X[0;0YFunctions for setting up an R-base[133X
      87.2-3 [33X[0;0YRefinement functions for the backtrack search[133X
      87.2-4 [33X[0;0YFunctions for meeting ordered partitions[133X
      87.2-5 [33X[0;0YAvoiding multiplication of permutations[133X
    87.3 [33X[0;0YStabilizer Chains for Automorphisms Acting on Enumerators[133X
      87.3-1 [33X[0;0YAn operation domain for automorphisms[133X
      87.3-2 [33X[0;0YEnumerators for cosets of characteristic factors[133X
      87.3-3 [33X[0;0YMaking automorphisms act on such enumerators[133X
  
  
  [32X
